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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2025 • Question 3

Question

Math 6420 • PDE Fall 2025 • Question 3

Let \(U\) be a bounded domain and \(u\) solve the heat equation \[ \begin {cases} \partial _t u=\Delta u+cu & \text {in } U\times (0,\infty ),\\ u(x,0)=g(x),\\ u(x,t)=0 & \text {on } \partial U\times (0,\infty ). \end {cases} \]

Classify, with justification, the optimal range of values for \(c\in \mathbb {R}\) so that for any smooth initial data \(g\) the solution \(u\) above converges to 0 as \(t\to \infty \).

Raw LaTeX

Let \(U\) be a bounded domain and \(u\) solve the heat equation
\[
\begin{cases}
\partial_t u=\Delta u+cu & \text{in } U\times(0,\infty),\\
u(x,0)=g(x),\\
u(x,t)=0 & \text{on } \partial U\times(0,\infty).
\end{cases}
\]

Classify, with justification, the optimal range of values for \(c\in\mathbb{R}\) so that for
any smooth initial data \(g\) the solution \(u\) above converges to 0 as \(t\to\infty\).

Topic Tags

Distributions and Weak Solutions Entropy Solutions, Shocks, and Rarefactions Fundamental Solutions Heat Equation and Parabolic Equations