let  be a bounded domain in rn , n=1,2,3 , with a smooth boundary . let  be an open subset of  with associated characteristic function . consider the following controlled linear fitzhugh  nagumo equation ( 1.1){utu+v+(x)u=gin(0,t),vt+vu=0in(0,t),u(t , x)=0on(0,t),u(0,x)=u0(x),v(0,x)=0in , where u is the electrical potential across the axonal membrane , v is a recovery variable , and g is the medicine actuator ( the control variable ) . for more details , see   and the references therein . further u0(x)c0()={yc():y=0on } is a given function , >0 and 0 are constants , c( ) and g is taken from the set of control functions : u{g:(0,+)rmeasurable:|g(t , x)|mfor  almost(t , x)(0,+) } , where m is a positive constant . solving the second equation of ( 1.1 ) in terms of u we get v=0te(ts)u(s)ds , and hence ( 1.1 ) can be equivalently expressed as the following integro - differential equation : ( 1.2){utu+0te(ts)u(s)ds+(x)u=gin(0,t),u(t , x)=0on(0,t),u(0,x)=u0(x)in. for each gu and t>0 , eq . ( 1.2 ) has a unique solution , denoted by uu(t , x;u0,g ) , in c([0,t];c0( ) ) ; see lemma  2.3 . in what follows , we shall omit the variables t and x for functions in ( t , x ) and we omit the variable x for functions in x , if there is no risk of causing confusion . the following time optimal control problem is the focus of this paper : ( p)inf{t:|u(t , x;u0,g)|1,x,gu}. without loss of generality we assume that u0( ) does not satisfy |u0(x)|1,x. we call tinf{t:|u(t , x;u0,g)|1,x,gu } the optimal ( minimal ) time for ( p ) and gu the associated time - optimal control with corresponding state u(t , x;u0,g ) , satisfying |u(t,x;u0,g)|1,x. we call a control gu an admissible control for ( p ) if there exists a t>0 such that |u(t , x;u0,g)|1,x. throughout this paper , for t>0 fixed , we write qt and t for the product sets ( 0,t) and ( 0,t) respectively . the lebesgue measure of a set d in rd(d1 ) is denoted by |d|rd . for every integer k0 , every p , r1,wk , p( ) , w01,p(),lr(0,t;w01,p()),c(qt ) and c0( ) stand for the usual sobolev s spaces , endowed with the usual norms . the main result of this paper is the following result which gives in maximum principle in ( 1.4 ) . theorem  1.1assume that 
 t 
 is the minimal time and let 
 g 
 be an optimal control for problem  ( p ) . then there exist 
 l65(0,t;w01,65( ) ) 
 and 
 0(c0()) 
 with 
 00 
 satisfying(1.3){t+tte(st)(s)ds(x)=0inqt,(t , x)=0ont,(t,)=0inand(1.4)(x)(t , x)g(t , x)=m|(x)(t , x)|a.e . assume that 
 t 
 is the minimal time and let 
 g 
 be an optimal control for problem  ( p ) . then there exist 
 l65(0,t;w01,65( ) ) 
 and 
 0(c0()) 
 with 
 00 
 satisfying(1.3){t+tte(st)(s)ds(x)=0inqt,(t , x)=0ont,(t,)=0inand(1.4)(x)(t , x)g(t , x)=m|(x)(t , x)|a.e . let us refer the selected results on pontryagin s maximum principle for time optimal control problems , governed by linear and semilinear parabolic differential equations . in but certainly , pointwise constraints in space and time are practically relevant . in  , pontryagin s maximum principle was considered for time optimal control problems governed by semilinear parabolic equations with pointwise constraints in space and time . however , to the best of our knowledge , pontryagin s maximum principle for time optimal control problems , governed by integro - differential equations with pointwise control constraint , has not been studied so far . based on theorem  1.1 , we obtain the bang corollary  1.2let 
 t 
 be the minimal time with an associated optimal control 
 g 
 for problem  ( p ) . if 
 0 
 in  theorem  1.1  satisfies0{l1(),forn=1,l2(),forn=2,3and(1.5)0(x)0a.e .  in  ,then|g(t , x)|=mfor almost  all(t , x)(0,t). let 
 t 
 be the minimal time with an associated optimal control 
 g 
 for problem  ( p ) . if 
 0 
 in  theorem  1.1  satisfies0{l1(),forn=1,l2(),forn=2,3and(1.5)0(x)0a.e .  in  ,then|g(t , x)|=mfor  almost  all(t , x)(0,t). the bang  bang property of the optimal controls is one of the main interests for time optimal control problems . for linear evolution equations this property with spatial integral constraints on the controls , pointwise in time , was first established in  . since then many results on the bang  bang property of optimal controls for time optimal control problems , governed by linear and semilinear parabolic differential equations , were obtained ( see e.g.  ) . in these papers , the  bang  bang  property of time optimal controls was proved for the boundary control problem for the heat equation , with pointwise control constraint and arbitrary reachable target set . here assumptions on the bounds to which the controls were subjected were utilized . in  , the bang bang property of optimal controls was established for time optimal control problems governed by linear parabolic equations , with pointwise control constraints . the target set was a point in the state space and the control acted globally onto the equation . to the best of our knowledge , the bang bang property for time optimal controls , acting locally onto parabolic and integro - differential equations with pointwise control constraints , has not been studied so far . from  , it is clear that there is an essential difference between the cases in which the control acts locally as opposed to globally onto the equations . bang property of the optimal controls for the heat equation can be derived by the similar arguments as used in this paper . for this case we believe that this result is also new for the heat equation . the rest of this paper is organized as follows . in section  2 , we provide existence results and a - priori estimates in the form that is required to obtain pontryagin s maximum principle for problem ( p ) . in section  3 , in this section , we present technical lemmas that will be essential for the proof of the maximum principle . first we introduce some notation . for p1 , the set { ylp(qt):yt , yxi , yxixjlp(qt),1i , jn } endowed with the norm ywp2,1(qt)=ylp(qt)+ytlp(qt)+i=1nyxilp(qt)+i , j=1nyxixjlp(qt ) is denoted by wp2,1(qt ) . for banach spaces x and y , we write x,,x,x and xy for the dual space of x , the duality pairing between x and x , and the product space endowed with the norm (,)xy=x+y , respectively . the following lemma is taken from   ( see proposition 3.4 and lemma 3.6 of chapter 4 in  ) . lemma  2.1let 
 x 
 be a banach space and let 
 s1 
 be finite codimensional in 
 x.(i)let 
 s2x 
 . then , for any 
 ar{0 } 
 and 
 br , the setas1bs2{as1bs2|s1s1,s2s2}is finite codimensional in 
 x.(ii)let 
 { fn}n1x 
 withfnx>0,fnfweakly  star  inx,andfn , sx,xn,ss1,n1,where 
  
 is a constant and 
 n0 let 
 x 
 be a banach space and let 
 s1 
 be finite codimensional in 
 x.(i)let 
 s2x 
 . then , for any 
 ar{0 } 
 and 
 br , the setas1bs2{as1bs2|s1s1,s2s2}is finite codimensional in 
 x.(ii)let 
 { fn}n1x 
 withfnx>0,fnfweakly  star  inx,andfn , sx,xn,ss1,n1,where 
  
 is a constant and 
 n0 then , for any 
 ar{0 } 
 and 
 br , the setas1bs2{as1bs2|s1s1,s2s2}is finite codimensional in 
 x. let 
 { fn}n1x 
 withfnx>0,fnfweakly  star  inx,andfn , sx,xn,ss1,n1,where 
  
 is a constant and 
 n0 
 . then 
 f0 . now we establish well - posedness of ( 1.2 ) in space of continuous functions . for this purpose , we introduce the operators a : d(a)c0()c0()c0()c0()andb : c0()c0()c0()c0( ) as follows : a(y , z)=(y,0),(y , z)d(a)=d()c0( ) , where d()={yc0():yc0( ) } , b(y , z)=(yz,yz),(y , z)c0()c0( ) . then we have the following lemma . lemma  2.2a+b 
 is the infinitesimal generator of an analytic semigroup , denoted by 
 t(t ) , on 
 c0()c0( ) . a+b 
 is the infinitesimal generator of an analytic semigroup , denoted by 
 t(t ) , on 
 c0()c0( ) . proofsince  is the infinitesimal generator of an analytic semigroup , denoted by t(t ) , on c0( ) ( see e.g. theorem 3.7 of chapter 7 in  ) , we get t(t)is  differentiable  fort>0 , and hence there exists a positive constant c such that ( see e.g. theorem 6.13 of chapter 2 in  ) ( 2.1)t(t)yc0()ct1yc0(),t>0,yc0( ) . for each t0 , we define the operator t(t ) from c0()c0( ) to c0()c0( ) by : t(t)(y , z)=(t(t)y , z),y , zc0( ) . it is obvious that t(t ) is a c0 semigroup with infinitesimal generator a , and it is differentiable for t>0 . moreover , by ( 2.1 ) , we have at(t)(y , z)c0()c0()ct1(y , z)c0()c0(),t>0,(y , z)c0()c0( ) , which , combined with theorem 5.2 of chapter 2 in  , shows that t(t ) is an analytic semigroup on c0()c0( ) . the result now follows since b is a bounded linear operator from c0()c0( ) to itself . since  is the infinitesimal generator of an analytic semigroup , denoted by t(t ) , on c0( ) ( see e.g. theorem 3.7 of chapter 7 in  ) , we get t(t)is  differentiable  fort>0 , and hence there exists a positive constant c such that ( see e.g. theorem 6.13 of chapter 2 in  ) ( 2.1)t(t)yc0()ct1yc0(),t>0,yc0( ) . for each t0 , we define the operator t(t ) from c0()c0( ) to c0()c0( ) by : t(t)(y , z)=(t(t)y , z),y , zc0( ) . it is obvious that t(t ) is a c0 semigroup with infinitesimal generator a , and it is differentiable for t>0 . moreover , by ( 2.1 ) , we have at(t)(y , z)c0()c0()ct1(y , z)c0()c0(),t>0,(y , z)c0()c0( ) , which , combined with theorem 5.2 of chapter 2 in  , shows that t(t ) is an analytic semigroup on c0()c0( ) . the result now follows since b is a bounded linear operator from c0()c0( ) to itself . then the equation{yty+0te(ts)y(s)ds+(x)y = ginqt , y(t , x)=0ont , y(0,x)=y0(x)inhas a unique solution 
 yc([0,t];c0( ) ) . then the equation{yty+0te(ts)y(s)ds+(x)y = ginqt , y(t , x)=0ont , y(0,x)=y0(x)inhas a unique solution 
 yc([0,t];c0( ) ) . proofthe existence and uniqueness of a solution in l2(0,t;h01())c([0,t],l2( ) ) can be argued by standard techniques  . let y1l2(0,t;h01( ) ) and y2c([0,t];h01( ) ) be the solutions to the following equations : ( 2.2){(y1)t=y10te(ts)y1(s)ds(x)y1inqt , y1(t , x)=0ont , y1(0,x)=y0(x)in and ( 2.3){(y2)t=y20te(ts)y2(s)ds(x)y2+ginqt , y2(t , x)=0ont , y2(0,x)=0in. it is obvious that ( 2.4)y = y1+y2inqt . ( 2.2 ) can be expressed as { ( y1)t=y1z1(x)y1inqt,(z1)t=z1+y1inqt , y1(t , x)=0ont , y1(0,x)=y0(x),z1(0,x)=0in , and hence from lemma  2.2 , it follows that ( 2.5)(y1,z1)=t()(y0(),0)c([0,t];c0())c([0,t];c0( ) ) . moreover , since y2c([0,t];h01( ) ) , it can be checked that 0te(ts)y2(s)dsy2+gl6(qt ) , which , combined with ( 2.3 ) and lp - theory for parabolic equation ( see e.g. theorem 1.14 of chapter 1 in  ) , establishes that y2w62,1(qt ) . since w62,1(qt ) embeds continuously into c(qt ) ( see e.g. theorem 1.4.1 in  ) and y2c([0,t];h01( ) ) , we infer that y2c([0,t];c0( ) ) . the existence and uniqueness of a solution in l2(0,t;h01())c([0,t],l2( ) ) can be argued by standard techniques  . let y1l2(0,t;h01( ) ) and y2c([0,t];h01( ) ) be the solutions to the following equations : ( 2.2){(y1)t=y10te(ts)y1(s)ds(x)y1inqt , y1(t , x)=0ont , y1(0,x)=y0(x)in and ( 2.3){(y2)t=y20te(ts)y2(s)ds(x)y2+ginqt , y2(t , x)=0ont , y2(0,x)=0in. it is obvious that ( 2.4)y = y1+y2inqt . ( 2.2 ) can be expressed as { ( y1)t=y1z1(x)y1inqt,(z1)t=z1+y1inqt , y1(t , x)=0ont , y1(0,x)=y0(x),z1(0,x)=0in , and hence from lemma  2.2 , it follows that ( 2.5)(y1,z1)=t()(y0(),0)c([0,t];c0())c([0,t];c0( ) ) . moreover , since y2c([0,t];h01( ) ) , it can be checked that 0te(ts)y2(s)dsy2+gl6(qt ) , which , combined with ( 2.3 ) and lp - theory for parabolic equation ( see e.g. theorem 1.14 of chapter 1 in  ) , establishes that y2w62,1(qt ) . since w62,1(qt ) embeds continuously into c(qt ) ( see e.g. theorem 1.4.1 in  ) and y2c([0,t];h01( ) ) , we infer that y2c([0,t];c0( ) ) . next we set u(qt)={g|qt : gu } and define the distance function d on u(qt ) by d(g , g)=|{(t , x)qt : g(t , x)g(t , x)}|rn+1,forg , gu(qt ) . by standard argument similar to that of proposition 3.10 , chapter 4 , in  , we have that ( u(qt),d ) is a complete metric space . the following lemma gives an approximation result for solutions u to ( 1.2 ) with respect to spike perturbations of the control g. lemma  2.4let 
 (0,1 ) 
 be a fixed constant . for every 
 g1,g2 
 in u(qt ) , there exists a measurable subset 
 eqt 
 such that 
 |e|rn+1=|qt|rn+1 
 . moreover , if we setg(t , x)={g1(t , x),(t , x)qte,g2(t , x),(t , x)e,then1(uu1)zc(qt)0as0,where 
 u 
 and 
 u1 
 are the solutions to  ( 1.2 )  corresponding to 
 g 
 and 
 g1 
 respectively , and 
 z 
 is the solution to(2.6){ztz+0te(ts)z(s)ds+(x)z=(g2g1)inqt , z(t , x)=0ont , z(0,x)=0in. let 
 (0,1 ) 
 be a fixed constant . for every 
 g1,g2 
 in 
 u(qt ) , there exists a measurable subset 
 eqt 
 such that 
 |e|rn+1=|qt|rn+1 
 . moreover , if we setg(t , x)={g1(t , x),(t , x)qte,g2(t , x),(t , x)e,then1(uu1)zc(qt)0as0,where 
 u 
 and 
 u1 
 are the solutions to  ( 1.2 )  corresponding to 
 g 
 and 
 g1 
 respectively , and 
 z 
 is the solution to(2.6){ztz+0te(ts)z(s)ds+(x)z=(g2g1)inqt , z(t , x)=0ont , z(0,x)=0in. proofby arguments similar to those for lemma 4.2 in  , there exists a sequence of measurable subsets { en}n=1 in qt , such that |en|rn+1=|qt|rn+1and1[en]1weakly  star  inl(qt)asn , where [en ] is the characteristic function of the set en . for the sake of completeness the proof is given in lemma  a.1 of the appendix . we set zn=1(unu1 ) , gn(t , x)={g1(t , x),(t , x)qten , g2(t , x),(t , x)en , and define by un(t , x ) the solution of ( 1.2 ) corresponding to gn . it is easy to check that ( 2.8){(zn)tzn=0te(ts)zn(s)ds(x)zn+1[en](g2g1)inqt , zn(t , x)=0ont , zn(0,x)=0in. from ( 2.8 ) it follows that ( 2.9)znc([0,t];h01())c1[en](g2g1)l2(qt ) . here and throughout this proof c denotes several positive constants independent of n. by ( 2.9 ) , we obtain 1[en](g2g1)0te(ts)zn(s)dsznl6(qt)c1[en](g2g1)l6(qt ) , which , combined with ( 2.8 ) and lp - theory for parabolic equations ( see e.g. theorem 1.14 of chapter 1 in  ) and ( 2.7 ) , shows that ( 2.10)znw62,1(qt)c1[en](g2g1)0te(ts)zn(s)dsznl6(qt)c1[en](g2g1)l6(qt)c . this , together with the fact that w62,1(qt ) embeds compactly in c(qt ) ( see e.g. theorem 1.4.1 in  ) , implies that there exist a subsequence denoted by { nk()}k1 , and a function zw62,1(qt ) , such that as k , ( 2.11)znk()zweakly  inw62,1(qt)andstrongly  inc(qt ) . passing to the limit k in ( 2.8 ) and by ( 2.7 ) with n = nk( ) , we see that z is the solution to eq . moreover , by ( 2.11 ) we obtain that there exists a positive integer nk0( ) satisfying 1(unk0()u1)zc(qt)=znk0()zc(qt).finally , we set e=enk0( ) and u=unk0( ) . by arguments similar to those for lemma 4.2 in  , there exists a sequence of measurable subsets { en}n=1 in qt , such that |en|rn+1=|qt|rn+1and1[en]1weakly  star  inl(qt)asn , where [en ] is the characteristic function of the set en . for the sake of completeness we set zn=1(unu1 ) , gn(t , x)={g1(t , x),(t , x)qten , g2(t , x),(t , x)en , and define by un(t , x ) the solution of ( 1.2 ) corresponding to gn . it is easy to check that ( 2.8){(zn)tzn=0te(ts)zn(s)ds(x)zn+1[en](g2g1)inqt , zn(t , x)=0ont , zn(0,x)=0in. from ( 2.8 ) it follows that ( 2.9)znc([0,t];h01())c1[en](g2g1)l2(qt ) . here and throughout this proof c denotes several positive constants independent of n. by ( 2.9 ) , we obtain 1[en](g2g1)0te(ts)zn(s)dsznl6(qt)c1[en](g2g1)l6(qt ) , which , combined with ( 2.8 ) and lp - theory for parabolic equations ( see e.g. theorem 1.14 of chapter 1 in  ) and ( 2.7 ) , shows that ( 2.10)znw62,1(qt)c1[en](g2g1)0te(ts)zn(s)dsznl6(qt)c1[en](g2g1)l6(qt)c . this , together with the fact that w62,1(qt ) embeds compactly in c(qt ) ( see e.g. theorem 1.4.1 in  ) , implies that there exist a subsequence denoted by { nk()}k1 , and a function zw62,1(qt ) , such that as k , ( 2.11)znk()zweakly  inw62,1(qt)andstrongly  inc(qt ) . passing to the limit k in ( 2.8 ) and by ( 2.7 ) with n = nk( ) , we see that z is the solution to eq . moreover , by ( 2.11 ) we obtain that there exists a positive integer nk0( ) satisfying 1(unk0()u1)zc(qt)=znk0()zc(qt). finally , we set e=enk0( ) and u=unk0( ) . the following lemma is concerned with an energy estimate , which will be used later . ,n ) 
 with 
 il6(qt),1in 
 . then the solution to(2.12){yty+0te(ts)y(s)ds+(x)y = fdivinqt , y(t , x)=0ont , y(0,x)=0insatisfies(2.13)yc(qt)c(fl6(qt)+i=1nil6(qt)),for a constant 
 c 
 independent of 
 f 
 and 
 . let 
 fl6(qt ) 
 and 
 =(1, to(2.12){yty+0te(ts)y(s)ds+(x)y = fdivinqt , y(t , x)=0ont , y(0,x)=0insatisfies(2.13)yc(qt)c(fl6(qt)+i=1nil6(qt)),for a constant 
 c 
 independent of 
 f 
 and 
 . proofby a density argument , it suffices to show that ( 2.13 ) holds for the case ic0(qt),1in . let y and y denote the solution to ( 2.12 ) with =0 , respectively with f=0 . then we have y = y+y.step  1 : =0. in this case , by the same arguments that were used to obtain ( 2.10 ) and continuous embedding of w62,1(qt ) into c(qt ) we have that ( 2.14)yc(qt)cyw62,1(qt)cfl6(qt ) . here and throughout the proof c denotes a generic constant independent of f and .step  2 : f=0 . we consider ( 2.15){yty=div0te(ts)y(s)dsinqt , y(t , x)=0ont , y(0,x)=0.in. multiplying both sides of the first equation by y , we obtain after integration on ( 0,t)y(t,)l2()2+0tyl2()2dsc0tl2()2ds+c0tyl2()2ds,t[0,t ] , which , combined with gronwall s inequality implies that ( 2.16)yc([0,t];l2())2+0tyl2()2dtc0tl2()2dt . next , for any t[0,t ] , we consider the equation ( 2.17){u(t)=0te(ts)y(s)dsin,u(t)=0on. from ( 2.17 ) and ( 2.16 ) it follows that ( 2.18)u(t,)h2()h01()2c0te(ts)y(s)dsl2()2c0tyl2()2dsc0tl2()2dt,t[0,t ] . by ( 2.17 ) we can rewrite ( 2.15 ) as : { yty=div(+u)inqt , y(t , x)=0ont , y(0,x)=0in. for any c0( ) we denote by zc(qt ) the solution to { ztz=0inqt , z(t , x)=0ont , z(0,x)=in. from these two equations we obtain ( 2.19)(x)y(t , x)dx=0t(z(t,x)y(,x)dx)d=0t[zt(t,x)y(,x)+z(t,x)y(,x)]dxd=0tz(t,x)div(+u)(,x)dxd=0tz(t,x)(+u)(,x)dxd,t[0,t ] . next we shall prove that ( 2.20)yc(qt)cl6(qt ) . for this purpose we define two operators a1:d(a1)l1()l1( ) and a65:d(a65)l65()l65( ) as follows : a1y=yforyd(a1),d(a1)={yw01,1(),yl1( ) } , where y is understood in the sense of distributions , and a65y=yforyd(a65),d(a65)=w2,65()w01,65( ) . then a1 and a65 are the infinitesimal generators of analytic semigroups on l1( ) and l65( ) respectively ( see e.g. theorems 3.10 and 3.5 of chapter 7 in  ) . let s1(t ) be the analytic semigroup generated by a1 . by theorem 4.3 of chapter 8 and theorem 6.13 of chapter 2 in   we have ( 2.21)z(t,)w1,65()c(a65)815z(t,)l65()=c(a1)815s1(t)l65()=c(a1)815s1(t2)s1(t2)l65()=cs1(t2)(a1)815s1(t2)l65( ) . moreover , by the proof of lemma 1 in   ( for the sake of completeness we provide this as lemma  a.2 in the appendix ) , we obtain s1(t2)(a1)815s1(t2)l65()tn12(a1)815s1(t2)l1( ) , which , together with ( 2.21 ) and theorem 6.13 of chapter 2 in   leads to z(t,)w1,65()ctn12(a1)815s1(t2)l1()ct815n12l1( ) . the latter inequality combined with ( 2.19 ) implies ( 2.22)|(x)y(t , x)dx|0tz(t,)l65()+ul6()dc0t(t)815n12l1()+ul6()dcl1()(0t+ul6()6d)16(0t(t)65(815n12)d)56cl1()+ul6(qt)[t165(815+n12)165(815+n12)]56cl1()+ul6(qt),t[0,t ] . hence , by ( 2.22 ) and ( 2.18 ) , we have |(x)y(t , x)dx|cl1()l6(qt),t[0,t ] , which implies that y(t,)l()cl6(qt),t[0,t ] . recalling the fact that yc(qt ) , this estimate implies ( 2.20).finally , it follows from ( 2.12 ) and ( 2.15 ) that yy satisfies { ( yy)t(yy)+0te(ts)(yy)ds+(x)(yy)=(x)yinqt,(yy)(t , x)=0ont,(yy)(0,x)=0in. by the estimate of step 1 and ( 2.20 ) we conclude that yc(qt)yyc(qt)+yc(qt)cyl6(qt)+yc(qt)cl6(qt ) , which together with ( 2.14 ) completes the proof . by a density argument , it suffices to show that ( 2.13 ) holds for the case ic0(qt),1in . let y and y denote the solution to ( 2.12 ) with =0 , respectively with f=0 . then we have y = y+y. step  1 : =0. in this case , by the same arguments that were used to obtain ( 2.10 ) and continuous embedding of w62,1(qt ) into c(qt ) we have that ( 2.14)yc(qt)cyw62,1(qt)cfl6(qt ) . here and throughout the proof c denotes a generic constant independent of f and . step  2 : f=0 . we consider ( 2.15){yty=div0te(ts)y(s)dsinqt , y(t , x)=0ont , y(0,x)=0.in. multiplying both sides of the first equation by y , we obtain after integration on ( 0,t)y(t,)l2()2+0tyl2()2dsc0tl2()2ds+c0tyl2()2ds,t[0,t ] , which , combined with gronwall s inequality implies that ( 2.16)yc([0,t];l2())2+0tyl2()2dtc0tl2()2dt . next , for any t[0,t ] , we consider the equation ( 2.17){u(t)=0te(ts)y(s)dsin,u(t)=0on. from ( 2.17 ) and ( 2.16 ) it follows that ( 2.18)u(t,)h2()h01()2c0te(ts)y(s)dsl2()2c0tyl2()2dsc0tl2()2dt,t[0,t ] . by ( 2.17 ) we can rewrite ( 2.15 ) as : { yty=div(+u)inqt , y(t , x)=0ont , y(0,x)=0in. for any c0( ) we denote by zc(qt ) the solution to { ztz=0inqt , z(t , x)=0ont , z(0,x)=in. from these two equations we obtain ( 2.19)(x)y(t , x)dx=0t(z(t,x)y(,x)dx)d=0t[zt(t,x)y(,x)+z(t,x)y(,x)]dxd=0tz(t,x)div(+u)(,x)dxd=0tz(t,x)(+u)(,x)dxd,t[0,t ] . for this purpose we define two operators a1:d(a1)l1()l1( ) and a65:d(a65)l65()l65( ) as follows : a1y=yforyd(a1),d(a1)={yw01,1(),yl1( ) } , where y is understood in the sense of distributions , and a65y=yforyd(a65),d(a65)=w2,65()w01,65( ) . then a1 and a65 are the infinitesimal generators of analytic semigroups on l1( ) and l65( ) respectively ( see e.g. theorems 3.10 and 3.5 of chapter 7 in  ) . let s1(t ) be the analytic semigroup generated by a1 . by theorem 4.3 of chapter 8 and theorem 6.13 of chapter 2 in   we have ( 2.21)z(t,)w1,65()c(a65)815z(t,)l65()=c(a1)815s1(t)l65()=c(a1)815s1(t2)s1(t2)l65()=cs1(t2)(a1)815s1(t2)l65( ) . moreover , by the proof of lemma 1 in   ( for the sake of completeness we provide this as lemma  a.2 in the appendix ) , we obtain s1(t2)(a1)815s1(t2)l65()tn12(a1)815s1(t2)l1( ) , which , together with ( 2.21 ) and theorem 6.13 of chapter 2 in   leads to z(t,)w1,65()ctn12(a1)815s1(t2)l1()ct815n12l1( ) . the latter inequality combined with ( 2.19 ) implies ( 2.22)|(x)y(t , x)dx|0tz(t,)l65()+ul6()dc0t(t)815n12l1()+ul6()dcl1()(0t+ul6()6d)16(0t(t)65(815n12)d)56cl1()+ul6(qt)[t165(815+n12)165(815+n12)]56cl1()+ul6(qt),t[0,t ] . hence , by ( 2.22 ) and ( 2.18 ) , we have |(x)y(t , x)dx|cl1()l6(qt),t[0,t ] , which implies that y(t,)l()cl6(qt),t[0,t ] . finally , it follows from ( 2.12 ) and ( 2.15 ) that yy satisfies { ( yy)t(yy)+0te(ts)(yy)ds+(x)(yy)=(x)yinqt,(yy)(t , x)=0ont,(yy)(0,x)=0in. by the estimate of step 1 and ( 2.20 ) we conclude that yc(qt)yyc(qt)+yc(qt)cyl6(qt)+yc(qt)cl6(qt ) , which together with ( 2.14 ) completes the proof . in the rest of this section , we shall consider the following adjoint equation ( 2.23){pt+ptte(st)p(s)ds(x)p=0inqt , p(t , x)=0ont , p(t,)=,in , where (c0()). definition  2.6a function pl65(qt ) is called a weak solution of ( 2.23 ) if qtpfdxdt=,y(t,)(c0()),c0(),fl6(qt ) , where y is the solution to ( 2.24){yty+0te(ts)y(s)ds+(x)y = finqt , y(t , x)=0ont , y(0,x)=0in. a function pl65(qt ) is called a weak solution of ( 2.23 ) if qtpfdxdt=,y(t,)(c0()),c0(),fl6(qt ) , where y is the solution to ( 2.24){yty+0te(ts)y(s)ds+(x)y = finqt , y(t , x)=0ont , y(0,x)=0in. lemma  2.7eq . ( 2.23 )  has a unique weak solution 
 p 
 . moreover , 
 pl65(0,t;w01,65( ) ) . proofwe choose a sequence { ln}n1c0( ) satisfying ( 2.25)lnl1()c(c0())andlndx,(c0()),c0(),c0( ) . here and throughout the proof of this lemma , c denotes a generic positive constant independent of n. for n=1 ,  , let pn be the solution to ( 2.26){(pn)tpn+tte(st)pn(s)ds+(x)pn=0inqt , pn(t , x)=0ont , pn(t , x)=ln(x)in.to obtain a bound for { pn } let =(f,)l6(qt)(l6(qt))n be arbitrary and denote by y the solution to ( 2.27){(y)ty+0te(ts)y(s)ds+(x)y=fdivinqt , y(t , x)=0ont , y(0,x)=0in.by ( 2.27 ) and ( 2.26 ) , we have qt[fpn+i=1ni(pn)xi]dxdt=y(t , x)pn(t , x)dx=y(t , x)ln(x)dx , which , together with ( 2.27 ) , lemma  2.5 and ( 2.25 ) , implies that |qt[fpn+i=1ni(pn)xi]dxdt|lnl1()y(t,)l()c(c0())(fl6(qt)+i=1nil6(qt ) ) . exist a subsequence of { pn}n=1 , still denoted by the same notation , and a function pl65(0,t;w01,65( ) ) such that ( 2.28)pnpweakly  inl65(0,t;w01,65( ) ) . next we claim that pis a weak solution to ( 2.23 )  . for this purpose , let fl6(qt ) and denote by y the solution to ( 2.24 ) . then by ( 2.24 ) and ( 2.26 ) , we get ( 2.29)qtpnfdxdt=ln(x)y(t , x)dx . passing to the limit for n in ( 2.29 ) we conclude by ( 2.28 ) , lemma  2.3 and ( 2.25 ) that p is a weak solution of ( 2.23).finally , we prove uniqueness . ( 2.23 ) . then p1p2l65(qt ) and ( 2.30)qt(p1p2)fdxdt=0,fl6(qt ) , which implies that p1p2=0 a.e . in qt and completes the proof . here and throughout the proof of this lemma , c denotes a generic positive constant independent of n. for n=1 ,  , let pn be the solution to ( 2.26){(pn)tpn+tte(st)pn(s)ds+(x)pn=0inqt , pn(t , x)=0ont , pn(t , x)=ln(x)in. to obtain a bound for { pn } let =(f,)l6(qt)(l6(qt))n be arbitrary and denote by y the solution to ( 2.27){(y)ty+0te(ts)y(s)ds+(x)y=fdivinqt , y(t , x)=0ont , y(0,x)=0in. by ( 2.27 ) and ( 2.26 ) , we have qt[fpn+i=1ni(pn)xi]dxdt=y(t , x)pn(t , x)dx=y(t , x)ln(x)dx , which , together with ( 2.27 ) , lemma  2.5 and ( 2.25 ) , implies that |qt[fpn+i=1ni(pn)xi]dxdt|lnl1()y(t,)l()c(c0())(fl6(qt)+i=1nil6(qt ) ) . exist a subsequence of { pn}n=1 , still denoted by the same notation , and a function pl65(0,t;w01,65( ) ) such that ( 2.28)pnpweakly  inl65(0,t;w01,65( ) ) . next we claim that pis a weak solution to ( 2.23 )  . for this purpose , let fl6(qt ) and denote by y the solution to ( 2.24 ) . then by ( 2.24 ) and ( 2.26 ) , passing to the limit for n in ( 2.29 ) we conclude by ( 2.28 ) , lemma  2.3 and ( 2.25 ) that p is a weak solution of ( 2.23 ) . finally , we prove uniqueness . ( 2.23 ) . then p1p2l65(qt ) and ( 2.30)qt(p1p2)fdxdt=0,fl6(qt ) , which implies that p1p2=0 a.e . in qt and completes the proof . introduction of a penalty functional 
 j:(u(qt);d)[0,+).let t > t be arbitrary and set s={sc0():sc0()1}. for any  with 0<<t , we define the functional j:(u(qt);d)[0,+ ) by : ( 3.1)j(g)=ds(u(t,;u0,g)),forgu(qt ) , where ds(u(t,;u0,g))infssu(t,;u0,g)sc0( ) . by lemma  2.5 , we have ( 3.2)|j(g)j(g)|u(t,;u0,g)u(t,;u0,g)c0()u(,;u0,g)u(,;u0,g)c(qt)cggl6(qt ) . here and throughout this proof , c denotes a generic constant independent of . from ( 3.2 ) , we can easily check that j is continuous on ( u(qt);d ) . moreover , it is obvious that j(g)>0 for each gu(qt ) . application of ekeland s variational principleby ekeland s variational principle , see e.g. corollary 2.2 of chapter 4 in  , we see that there exists a gu(qt ) such that ( 3.3)d(g,g)[()]12 and ( 3.4)[()]12d(g,g)j(g)j(g),gu(qt).step  3 . derivation of the necessary conditions for 
 ( g,u ) , where 
 u(t , x)u(t , x;u0,g).let gu(qt ) be arbitrary . then by lemma  2.4 there exists for each (0,1 ) a measurable set eqt such that |e|rn+1=|qt|rn+1 , and the function ( 3.5)g(t , x){g(t , x),(t , x)qte,g(t , x),(t , x)e satisfies gu(qt ) . moreover , the solution u(t , x)u(t , x;u0,g ) satisfies : ( 3.6)1(uu)zc(qt)0as0 , where z is the unique solution to ( 3.7){(z)tz+0te(ts)z(s)ds+(x)z=(gg)inqt , z(t , x)=0ont , z(0,x)=0in. from ( 3.4 ) and ( 3.5 ) it follows that [()]12|qt|rn+1=[()]12d(g,g)j(g)j(g ) , which , together with ( 3.1 ) and ( 3.6 ) , implies ( 3.8)[()]12|qt|rn+1j(g)j(g)=ds(u(t,))ds(u(t,)),z(t,)(c0()),c0()as0 + , see e.g. proposition 3.11 of chapter 4 in  . here since t is the minimal time for problem ( p ) , we have u(t,)s and hence ( 3.9)(c0())=1,for  each  >0 , see e.g. proposition 3.11 of chapter 4 in  .step  4 . passing to the limit for 
 0 
 in  ( 3.7 ) and ( 3.8).by ( 3.3 ) and using g,gu(qt ) , one can easily show that ( 3.10)ggstrongly  inl6(qt)as0 . making use of ( 1.2 ) satisfied by ( u,g ) and ( u,g ) , we obtain from lemma  2.5 that ( 3.11)uuc(qt)cggl6(qt)0as0 . by ( 3.10 ) , ( 3.7 ) and similar arguments as in ( 3.11 ) , we see that ( 3.12)zzc(qt)cggl6(qt)0as0 , where z is the unique solution to ( 3.13){ztz+0te(ts)z(s)ds+(x)z=(gg)inqt , z(t , x)=0ont , z(0,x)=0in. moreover , by ( 3.8 ) and ( 3.9 ) , we get that ( 3.14),z(t,)(c0()),c0()=,z(t,)z(t,)(c0()),c0()+,z(t,)(c0()),c0()z(t,)z(t,)c0()[()]12|qt|rn+1 . applying ( 3.9 ) again , we can assume , without loss of generality , that ( 3.15)0weakly  star  in(c0()) , for some 0(c0()). it follows easily from ( 3.14 ) , ( 3.15 ) and ( 3.12 ) that ( 3.16)0,z(t,)(c0()),c0()0.step  5 . the proof of  ( 1.3 ) and ( 1.4).we first claim that ( 3.17)00 . indeed , by ( 3.9 ) and making use of the definition of the subdifferential ds , we obtain ,u(t,)w(c0()),c0()=,u(t,)u(t,)(c0()),c0()+,u(t,)w(c0()),c0()u(t,)u(t,)c0()+ds(u(t,))u(t,)u(t,)c0( ) , which combined with ( 3.11 ) implies that ,u(t,)w(c0()),c0()u(t,)u(t,)c0()0,ws , as  0 . utilizing this relation , together with lemma  2.1 , ( 3.9 ) , ( 3.15 ) and the fact that s is finite codimensional in c0( ) , ( 3.17 ) follows.let l65(0,t;w01,65( ) ) be the unique weak solution to ( 1.3 ) . by the proof of lemma  2.7 , we infer that there exist two sequences { ln}n=1c0( ) and { pn}n=1 satisfying ( 3.18)ln(x)(x)dx0,(c0()),c0()c0(),(3.19)pnweakly  inl65(0,t;w01,65( ) ) and ( 3.20){(pn)t+pntte(st)pn(s)ds(x)pn=0inqt,pn(t , x)=0ont,pn(t,x)=ln(x)in. from ( 3.13 ) and ( 3.20 ) it follows that ln(),z(t,)l2(),l2()=pn(t,),z(t,)l2(),l2()=qt(gg)pndxdt . passing to the limit for n in the last equality , we obtain by ( 3.18 ) , ( 3.19 ) and ( 3.16 ) that qt(gg)dxdt0,gu(qt ) , from which we conclude ( 3.21)qt(gg)dxdt0,guad , where uad{g : qt(,+)measurable;|g(t , x)|mfor  almost(t , x)qt}. finally , we denote ( 3.22)f(t , x)=(x)(mg(t , x))(t , x),(t , x)qt , which satisfies fl1(qt ) . therefore , there exists a measurable set dqt with |d|rn+1=|qt|rn+1 , such that any point in d is a lebesgue point of f , i.e. ,  ( 3.23)limr0+|b((t,x),r)|rn+11b((t,x),r)|f(t , x)f(t,x)|dxdt=0,(t,x)d , where b((t,x),r ) denotes the closed ball with center at ( t,x ) and radius r in rn+1 . now for any fixed ( t,x)d , and for any sufficiently small positive constant r , we define ( 3.24)gr(t , x)={g(t , x),if(t , x)b((t,x),r)cqt,m , if(t , x)b((t,x),r)qt. here b((t,x),r)c denotes the complement to b((t,x),r ) . from ( 3.24 ) and ( 3.21 ) it follows that ( 3.25)b((t,x),r)qt(x)(mg(t , x))(t , x)dxdt0,r>0 . dividing ( 3.25 ) by |b((t,x),r)|rn+1 we obtain by ( 3.22 ) and ( 3.23)m(x)(t , x)g(t , x)(x)(t , x),(t , x)d . this inequality together with dqt and |d|rn+1=|qt|rn+1 implies that ( 3.26)m(x)(t , x)g(t , x)(x)(t , x),a.e.(t , x)qt. similarly , we obtain ( 3.27)m(x)(t , x)g(t , x)(x)(t , x),a.e.(t , x)qt. moreover , since max|a|m((x)(t , x)a)=maxa{m,m}((x)(t , x)a ) , we deduce the desired result from ( 3.26 ) and ( 3.27)(x)(t , x)g(t , x)=max|a|m((x)(t , x)a)=m|(x)(t , x)| .   the proof is split into five steps . let t > t be arbitrary and set s={sc0():sc0()1}. for any  with 0<<t , we define the functional j:(u(qt);d)[0,+ ) by : ( 3.1)j(g)=ds(u(t,;u0,g)),forgu(qt ) , where ds(u(t,;u0,g))infssu(t,;u0,g)sc0( ) . by lemma  2.5 , we have ( 3.2)|j(g)j(g)|u(t,;u0,g)u(t,;u0,g)c0()u(,;u0,g)u(,;u0,g)c(qt)cggl6(qt ) . here and throughout this proof , c denotes a generic constant independent of . from ( 3.2 ) , we can easily check that j is continuous on ( u(qt);d ) . moreover , it is obvious that j(g)>0 for each gu(qt ) . abbreviating u(t , x)=u(t , x;u0,g ) , we have that j(g)=ds(u(t,))()u(t,)u(t,)c0()0as0 . application of ekeland s variational principle by ekeland s variational principle , see e.g. corollary 2.2 of chapter 4 in  , we see that there exists a gu(qt ) such that ( 3.3)d(g,g)[()]12 and ( 3.4)[()]12d(g,g)j(g)j(g),gu(qt ) . step  3 . derivation of the necessary conditions for 
 ( g,u ) , where 
 u(t , x)u(t , x;u0,g ) . then by lemma  2.4 there exists for each (0,1 ) a measurable set eqt such that |e|rn+1=|qt|rn+1 , and the function ( 3.5)g(t , x){g(t , x),(t , x)qte,g(t , x),(t , x)e satisfies gu(qt ) . moreover , the solution u(t , x)u(t , x;u0,g ) satisfies : ( 3.6)1(uu)zc(qt)0as0 , where z is the unique solution to ( 3.7){(z)tz+0te(ts)z(s)ds+(x)z=(gg)inqt , z(t , x)=0ont , z(0,x)=0in. from ( 3.4 ) and ( 3.5 ) it follows that [()]12|qt|rn+1=[()]12d(g,g)j(g)j(g ) , which , together with ( 3.1 ) and ( 3.6 ) , implies ( 3.8)[()]12|qt|rn+1j(g)j(g)=ds(u(t,))ds(u(t,)),z(t,)(c0()),c0()as0 + , see e.g. proposition 3.11 of chapter 4 in  . here since t is the minimal time for problem ( p ) , we have u(t,)s and hence ( 3.9)(c0())=1,for  each  >0 , see e.g. proposition 3.11 of chapter 4 in  . step  4 . passing to the limit for 
 0 
 in  ( 3.7 ) and ( 3.8 ) . by ( 3.3 ) and using g,gu(qt ) , one can easily show that ( 3.10)ggstrongly  inl6(qt)as0 . making use of ( 1.2 ) satisfied by ( u,g ) and ( u,g ) , we obtain from lemma  2.5 that ( 3.11)uuc(qt)cggl6(qt)0as0 . by ( 3.10 ) , ( 3.7 ) and similar arguments as in ( 3.11 ) , we see that ( 3.12)zzc(qt)cggl6(qt)0as0 , where z is the unique solution to ( 3.13){ztz+0te(ts)z(s)ds+(x)z=(gg)inqt , z(t , x)=0ont , z(0,x)=0in. moreover , by ( 3.8 ) and ( 3.9 ) , we get that ( 3.14),z(t,)(c0()),c0()=,z(t,)z(t,)(c0()),c0()+,z(t,)(c0()),c0()z(t,)z(t,)c0()[()]12|qt|rn+1 . applying ( 3.9 ) again , we can assume , without loss of generality , that ( 3.15)0weakly  star  in(c0()) , for some 0(c0()). it follows easily from ( 3.14 ) , ( 3.15 ) and ( 3.12 ) that ( 3.16)0,z(t,)(c0()),c0()0 . we first claim that ( 3.17)00 . indeed , by ( 3.9 ) and making use of the definition of the subdifferential ds , we obtain ,u(t,)w(c0()),c0()=,u(t,)u(t,)(c0()),c0()+,u(t,)w(c0()),c0()u(t,)u(t,)c0()+ds(u(t,))u(t,)u(t,)c0( ) , which combined with ( 3.11 ) implies that ,u(t,)w(c0()),c0()u(t,)u(t,)c0()0,ws , as  0 . utilizing this relation , together with lemma  2.1 , ( 3.9 ) , ( 3.15 ) and the fact that s is finite codimensional in c0( ) , ( 3.17 ) follows . let l65(0,t;w01,65( ) ) be the unique weak solution to ( 1.3 ) . by the proof of lemma  2.7 , we infer that there exist two sequences { ln}n=1c0( ) and { pn}n=1 satisfying ( 3.18)ln(x)(x)dx0,(c0()),c0()c0(),(3.19)pnweakly  inl65(0,t;w01,65( ) ) and ( 3.20){(pn)t+pntte(st)pn(s)ds(x)pn=0inqt,pn(t , x)=0ont,pn(t,x)=ln(x)in. from ( 3.13 ) and ( 3.20 ) it follows that ln(),z(t,)l2(),l2()=pn(t,),z(t,)l2(),l2()=qt(gg)pndxdt . passing to the limit for n in the last equality , we obtain by ( 3.18 ) , ( 3.19 ) and ( 3.16 ) that qt(gg)dxdt0,gu(qt ) , from which we conclude ( 3.21)qt(gg)dxdt0,guad , where uad{g : qt(,+)measurable;|g(t , x)|mfor  almost(t , x)qt}. finally , we denote ( 3.22)f(t , x)=(x)(mg(t , x))(t , x),(t , x)qt , which satisfies fl1(qt ) . therefore , there exists a measurable set dqt with |d|rn+1=|qt|rn+1 , such that any point in d is a lebesgue point of f , i.e. ,  ( 3.23)limr0+|b((t,x),r)|rn+11b((t,x),r)|f(t , x)f(t,x)|dxdt=0,(t,x)d , where b((t,x),r ) denotes the closed ball with center at ( t,x ) and radius r in rn+1 . now for any fixed ( t,x)d , and for any sufficiently small positive constant r , we define ( 3.24)gr(t , x)={g(t , x),if(t , x)b((t,x),r)cqt,m , if(t , x)b((t,x),r)qt. here b((t,x),r)c denotes the complement to b((t,x),r ) . from ( 3.24 ) and ( 3.21 ) it follows that ( 3.25)b((t,x),r)qt(x)(mg(t , x))(t , x)dxdt0,r>0 . dividing ( 3.25 ) by |b((t,x),r)|rn+1 we obtain by ( 3.22 ) and ( 3.23)m(x)(t , x)g(t , x)(x)(t , x),(t , x)d . this inequality together with dqt and |d|rn+1=|qt|rn+1 implies that ( 3.26)m(x)(t , x)g(t , x)(x)(t , x),a.e.(t , x)qt. similarly , we obtain ( 3.27)m(x)(t , x)g(t , x)(x)(t , x),a.e.(t , x)qt. moreover , since max|a|m((x)(t , x)a)=maxa{m,m}((x)(t , x)a ) , we deduce the desired result from ( 3.26 ) and ( 3.27)(x)(t , x)g(t , x)=max|a|m((x)(t , x)a)=m|(x)(t , x)| .   we turn to the proof of  corollary  1.2 . prooflet (t , x)=(tt , x),(t , x)=0te(ts)(s , x)ds,(t , x)qt. then we have ( 3.28){t=(x)inqt,t=+inqt,(t , x)=0ont,(0,x)=0(x),(0,x)=0in. we shall carry out the proof in three steps.step  1 . there exists a sequence 
 { tn}n=1(0,t ) 
 with 
 tn0 
 such that(3.29)(tn , x)(0,x)for  a.e.x,and(3.30)t(t,x)is  analytic  on(0,t),x. this will be achieved for the cases n=1 and n=2,3 separately.case  1 : n=1.we define two operators a1:d(a1)l1()l1()l1()l1( ) and b1:l1()l1()l1()l1( ) as follows : a1(y , z)=(y,0),(y , z)d(a1)=d()l1( ) , where d()={yw01,1():yl1()},y is understood in the sense of distribution , and b1(y , z)=(yz,yz),(y , z)l1()l1( ) . let s1(t),t0 be the analytic semigroup , generated by  , on l1( ) ( see e.g. theorem 3.10 of chapter 7 in  ) . for t0 , we define s1(t):l1()l1()l1()l1( ) by : s1(t)(y , z)=(s1(t)y , z),(y , z)l1( ) . it is easy to check that s1(t ) is a c0-semigroup with infinitesimal generator a1. furthermore , since b1 is a linear bounded operator , a1+b1 is the infinitesimal generator of a c0-semigroup on l1()l1( ) , denoted by s1(t ) . this , combined with ( 3.28 ) implies that ( ,)=s1()(0,0)c([0,t];l1())c([0,t];l1( ) ) , from which ( 3.29 ) follows.next we turn to the proof of ( 3.30 ) . by theorem  1.1 since w01,65()c0( ) for n=1 these facts imply that for each t0>0 there exists t1(0,t0 ) , such that ( 3.31)(t1,)c0()andc([0,t];c0( ) ) . then by ( 3.28 ) and ( 3.31 ) , we have ( 3.32){(1)t=11(x)1inqtt1,(1)t=1+1inqtt1,1(t , x)=0ontt1,1(0,x)=(t1,x),1(0,x)=(t1,x)in. by lemma  2.2 and ( 3.31 ) we deduce that for any ( 1,2)(c0())(c0()) the function ( 3.33)t(1,2),t(t)((t1,),(t1,))(c0())(c0()),c0()c0()is  analytic  on(0, ) . for any x0 , we define ( 1,2)(c0())(c0()) by (1,2),(y , z)(c0())(c0()),c0()c0()=(y(x0),z(x0)),(y , z)c0()c0( ) , which , combined with ( 3.33 ) , implies that t(t(t)((t1,),(t1,)))(x0)is  analytic  on(0, ) . this together with ( 3.32 ) and the fact that 1(t , x)=(t+t1,x),(t , x)qtt1 , implies that t(t , x0)is  analytic  on(t1,t ) , and hence ( 3.30 ) follows.case  2 : n=2,3.by the same arguments as in case 1 we find that c([0,t];l2( ) ) , which implies ( 3.29 ) and ( 3.34)f0te(ts)(s)dsl2(qt ) . we can easily check that ( 3.35)=1+2 , where 1 and 2 are the solutions to { ( 1)t1=0inqt,1(t , x)=0ont,1(0,x)=0(x)in and ( 3.36){(2)t2=finqt,2(t , x)=0ont,2(0,x)=0in , respectively . since  is the infinitesimal generator of an analytic semigroup on l2( ) with domain d()=w2,2()w01,2( ) , we obtain by theorem 4.3 of chapter 8 and theorem 6.13 of chapter 2 in   that for n=2 , ( 3.37)1(t,)w01,3()c()561(t,)l2()ct560l2()ct56,t>0 , and for n=3 , ( 3.38)1(t,)w01,258()c()1792001(t,)l2()ct1792000l2()ct179200,t>0 . here c denotes a generic constant independent of t. from ( 3.37 ) , ( 3.38 ) and sobolev s embedding theorem it follows that ( 3.39)1{l109(0,t;w01,3())l109(0,t;c0()),forn=2,l109(0,t;w01,258())l109(0,t;c0()),forn=3 . moreover , by ( 3.34 ) , ( 3.36 ) and sobolev s embedding theorem , we have 2l2(0,t;w2,2()w01,2())l2(0,t;c0( ) ) . this together with ( 3.39 ) and ( 3.35 ) implies that l109(0,t;c0( ) ) . hence ( 3.31 ) follows . the proof of ( 3.30 ) can now be completed by the same arguments as in case 1.step  2 : there exists a measurable set 
  
 with 
 ||rn=||rn , such that(3.40)|{t(0,t):(t , x)=0}|r>0(0,x)=0,x. indeed , by ( 3.29 ) , we have that there exists a measurable set , denoted by  , such that ,||rn=||rn and ( 3.41)(tn , x)(0,x),x. for each x , if |{t(0,t):(t , x)=0}|r>0 , we obtain by ( 3.30 ) that (t , x)=0 for t(0,t ) . then by ( 3.41 ) , we complete the proof of ( 3.40).step  3 : ( 3.42)|{(t , x)qt:(t , x)=0}|rn+1t|{x:(0,x)=0}|rn . to verify this claim we set ={x:(0,x)0}. by ( 3.40 ) we get for x that |{t(0,t):(t , x)=0}|r=0 . hence there exists a set ex(0,t ) with |ex|r = t such that ( 3.43)(t , x)0,tex . by fubini s theorem , we have ( 3.44)|{(t , x)(0,t):(t , x)0}|rn+1=|sx|rdx , wheresx={t(0,t):(t , x)0}. from ( 3.43 ) it follows that exsx,x. hence |sx|r = t,x. this combined with ( 3.44 ) shows that |{(t , x)(0,t):(t , x)0}|rn+1|sx|rdx = t||rn = t|{x:(0,x)0}|rn . hence we have |{(t , x)(0,t):(t , x)=0}|rn+1=t||rn|{(t , x)(0,t):(t , x)0}|rn+1t||rnt|{x:(0,x)0}|rnt|{x:(0,x)=0}|rn , which , together with the fact that  and ||rn=||rn , implies ( 3.42).by ( 3.42 ) and ( 1.5 ) , we get |{(t , x)qt:(t , x)=0}|rn+1=0 , i.e. , (t , x)0 a.e . in qt. noticing that (t , x)=(tt , x ) in qt , we infer (t , x)0 a.e . in qt. this together with ( 1.4 ) completes the proof of the corollary . let (t , x)=(tt , x),(t , x)=0te(ts)(s , x)ds,(t , x)qt. then we have ( 3.28){t=(x)inqt,t=+inqt,(t , x)=0ont,(0,x)=0(x),(0,x)=0in. we shall carry out the proof in three steps . step  1 . there exists a sequence 
 { tn}n=1(0,t ) 
 with 
 tn0 
 such that(3.29)(tn , x)(0,x)for  a.e.x,and(3.30)t(t,x)is  analytic  on(0,t),x. this will be achieved for the cases n=1 and n=2,3 separately . we define two operators a1:d(a1)l1()l1()l1()l1( ) and b1:l1()l1()l1()l1( ) as follows : a1(y , z)=(y,0),(y , z)d(a1)=d()l1( ) , where d()={yw01,1():yl1()},y is understood in the sense of distribution , and b1(y , z)=(yz,yz),(y , z)l1()l1( ) . let s1(t),t0 be the analytic semigroup , generated by  , on l1( ) ( see e.g. theorem 3.10 of chapter 7 in  ) . for t0 , we define s1(t):l1()l1()l1()l1( ) by : s1(t)(y , z)=(s1(t)y , z),(y , z)l1( ) . it is easy to check that s1(t ) is a c0-semigroup with infinitesimal generator a1. furthermore , since b1 is a linear bounded operator , a1+b1 is the infinitesimal generator of a c0-semigroup on l1()l1( ) , denoted by s1(t ) . this , combined with ( 3.28 ) implies that ( ,)=s1()(0,0)c([0,t];l1())c([0,t];l1( ) ) , from which ( 3.29 ) follows . next we turn to the proof of ( 3.30 ) . by theorem  1.1 , we have l65(0,t;w01,65( ) ) , and hence c([0,t];w01,65( ) ) . since w01,65()c0( ) for n=1 these facts imply that for each t0>0 there exists t1(0,t0 ) , such that ( 3.31)(t1,)c0()andc([0,t];c0( ) ) . then by ( 3.28 ) and ( 3.31 ) , we have ( 3.32){(1)t=11(x)1inqtt1,(1)t=1+1inqtt1,1(t , x)=0ontt1,1(0,x)=(t1,x),1(0,x)=(t1,x)in. by lemma  2.2 and ( 3.31 ) we deduce that for any ( 1,2)(c0())(c0()) the function ( 3.33)t(1,2),t(t)((t1,),(t1,))(c0())(c0()),c0()c0()is  analytic  on(0, ) . for any x0 , we define ( 1,2)(c0())(c0()) by (1,2),(y , z)(c0())(c0()),c0()c0()=(y(x0),z(x0)),(y , z)c0()c0( ) , which , combined with ( 3.33 ) , implies that t(t(t)((t1,),(t1,)))(x0)is  analytic  on(0, ) . this together with ( 3.32 ) and the fact that 1(t , x)=(t+t1,x),(t , x)qtt1 , implies that t(t , x0)is  analytic  on(t1,t ) , and hence ( 3.30 ) follows . by the same arguments as in case 1 we find that c([0,t];l2( ) ) , which implies ( 3.29 ) and ( 3.34)f0te(ts)(s)dsl2(qt ) . we can easily check that ( 3.35)=1+2 , where 1 and 2 are the solutions to { ( 1)t1=0inqt,1(t , x)=0ont,1(0,x)=0(x)in and ( 3.36){(2)t2=finqt,2(t , x)=0ont,2(0,x)=0in , respectively . since  is the infinitesimal generator of an analytic semigroup on l2( ) with domain d()=w2,2()w01,2( ) , we obtain by theorem 4.3 of chapter 8 and theorem 6.13 of chapter 2 in   that for n=2 , ( 3.37)1(t,)w01,3()c()561(t,)l2()ct560l2()ct56,t>0 , and for n=3 , ( 3.38)1(t,)w01,258()c()1792001(t,)l2()ct1792000l2()ct179200,t>0 . here c denotes a generic constant independent of t. from ( 3.37 ) , ( 3.38 ) and sobolev s embedding theorem it follows that ( 3.39)1{l109(0,t;w01,3())l109(0,t;c0()),forn=2,l109(0,t;w01,258())l109(0,t;c0()),forn=3 . moreover , by ( 3.34 ) , ( 3.36 ) and sobolev s embedding theorem , we have 2l2(0,t;w2,2()w01,2())l2(0,t;c0( ) ) . this together with ( 3.39 ) and ( 3.35 ) implies that l109(0,t;c0( ) ) . hence ( 3.31 ) follows . the proof of ( 3.30 ) can now be completed by the same arguments as in case 1 . step  2 : there exists a measurable set 
  
 with 
 ||rn=||rn , such that(3.40)|{t(0,t):(t , x)=0}|r>0(0,x)=0,x. indeed , by ( 3.29 ) , we have that there exists a measurable set , denoted by  , such that ,||rn=||rn and ( 3.41)(tn , x)(0,x),x. for each x , if |{t(0,t):(t , x)=0}|r>0 , we obtain by ( 3.30 ) that (t , x)=0 for t(0,t ) . step  3 : ( 3.42)|{(t , x)qt:(t , x)=0}|rn+1t|{x:(0,x)=0}|rn . to verify this claim we set ={x:(0,x)0}. by ( 3.40 ) hence there exists a set ex(0,t ) with |ex|r = t such that ( 3.43)(t , x)0,tex . by fubini s theorem , we have ( 3.44)|{(t , x)(0,t):(t , x)0}|rn+1=|sx|rdx , wheresx={t(0,t):(t , x)0}. from ( 3.43 ) it follows that exsx,x. hence |sx|r = t,x. this combined with ( 3.44 ) shows that |{(t , x)(0,t):(t , x)0}|rn+1|sx|rdx = t||rn = t|{x:(0,x)0}|rn . hence we have |{(t , x)(0,t):(t , x)=0}|rn+1=t||rn|{(t , x)(0,t):(t , x)0}|rn+1t||rnt|{x:(0,x)0}|rnt|{x:(0,x)=0}|rn , which , together with the fact that  and ||rn=||rn , implies ( 3.42 ) . by ( 3.42 ) and ( 1.5 ) , we get |{(t , x)qt:(t , x)=0}|rn+1=0 , i.e. , (t , x)0 a.e . in qt. noticing that (t , x)=(tt , x ) in qt , we infer (t , x)0 a.e . in qt. this together with ( 1.4 ) completes the proof of the corollary . remark  3.1for the controlled heat equation { utu=ginqt , u(t , x)=0ont , u(0,x)=u0(x)in , where  is a bounded domain in rn , n1 , with a sufficiently smooth boundary  , consider the time optimal control problem with the same pointwise control constraint as problem ( p ) . bang property for time optimal controls without assumptions on 0 as in corollary  1.2 can be obtained . indeed , in this case , from the proof of theorem  1.1 , pontryagin s maximum principle is obtained in the form : there exist l1(0,t;w01,1( ) ) and 0(c0()) with 00 satisfying { t+=0inqt,(t , x)=0ont,(t,)=0in and ( 3.45)(x)(t , x)g(t , x)=m|(x)(t , x)|a.e . inqt. by theorem 4.7.12 in  , the smoothing effect of the heat equation and the fact that 00 , we infer that (t , x)0a.e.(t , x)qt. this together with ( 3.45 ) implies that |g(t , x)|=ma.e . in(0,t). this result can be extended to the case where the laplacian is replaced by lu(x)=i , j=1n(aij(x)uxi(x))xj+i=1nbi(x)uxi(x)+c(x)u(x ) , where aij(x)=aji(x ) and l is uniformly elliptic . for the controlled heat equation { utu=ginqt , u(t , x)=0ont , u(0,x)=u0(x)in , where  is a bounded domain in rn , n1 , with a sufficiently smooth boundary  , consider the time optimal control problem with the same pointwise control constraint as problem ( p ) bang property for time optimal controls without assumptions on 0 as in corollary  1.2 can be obtained . indeed , in this case , from the proof of theorem  1.1 , pontryagin s maximum principle is obtained in the form : there exist l1(0,t;w01,1( ) ) and 0(c0()) with 00 satisfying { t+=0inqt,(t , x)=0ont,(t,)=0in and ( 3.45)(x)(t , x)g(t , x)=m|(x)(t , x)|a.e .  inqt. by theorem 4.7.12 in  , the smoothing effect of the heat equation and the fact that 00 , we infer that (t , x)0a.e.(t , x)qt. this together with ( 3.45 ) implies that |g(t , x)|=ma.e . in(0,t). this result can be extended to the case where the laplacian is replaced by lu(x)=i , j=1n(aij(x)uxi(x))xj+i=1nbi(x)uxi(x)+c(x)u(x ) , where aij(x)=aji(x ) and l is uniformly elliptic .
time optimal control governed by the internally controlled linear fitzhugh  nagumo equation with pointwise control constraint is considered . making use of ekeland s variational principle 
 , we obtain pontryagin s maximum principle for a time optimal control problem . using the maximum principle , the bang  bang property of the optimal controls 
 is established under appropriate assumptions .