Let $u_0 \in L^2(\mathbb{R}^n)$ and define $u(t,\cdot)$ for $t>0$ by convolution with the heat kernel. Then $u(t,\cdot)\in C^\infty(\mathbb{R}^n)$ for every $t>0$ and $u$ solves $\partial_t u=\Delta u$ on $(0,\infty)\times\mathbb{R}^n$.
Knowledge Status
Analysis
Discussion
A heat-kernel smoothing statement asserting that convolution with the heat kernel turns L^2 initial data into smooth positive-time solutions.
Proof
No proof available for this theorem.
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Prerequisites Graph
Interactive dependency map showing how this theorem builds on foundational concepts