+1 vote
485 views
Let $u$ be the unique solution of

$$\left.\begin{array}{lr}\dfrac{\partial u}{\partial t}=\dfrac{\partial^{2} u}{\partial x^{2}} \text { where }(x, t) \in(0,1) \times(0, \infty) \\ u(x, 0)=\sin \pi x, & x \in(0,1) \\ u(0, t)=u(1, t)=0, & t \in(0, \infty)\end{array}\right\}$$

Then which of the following is true?

1. There exists $(x, t) \in(0,1) \times(0, \infty)$ such that $u(x, t)=0$

2. There exists $(x, t) \in(0,1) \times(0, \infty)$ such that $\frac{\partial u}{\partial t}(x, t)=0$

3. The function $e^{t} u(x, t)$ is bounded for $(x, t) \in(0,1) \times(0, \infty)$

4. There exists $(x, t) \in(0,1) \times(0, \infty)$ such that $u(x, t)>1$
| 485 views
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.978224515914917, "perplexity": 363.0457940936322}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500250.51/warc/CC-MAIN-20230205063441-20230205093441-00567.warc.gz"}