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Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.

1 vote
0 answers
777 views

$C^{1,2}$ regularity of (weak) solutions to the heat equation

Let $\Omega$ be a bounded Lipschitz domain (smoother if needed), and consider the heat equation $$u_t - \Delta u = 0$$ $$\frac{\partial u(t,x)}{\partial \nu(x)} = a(t,x) - b(t,x)u(t,x)$$ $$u(0) = u_0$ …
ChristopherSail's user avatar
2 votes
1 answer
1k views

Strong maximum principle for heat equation. Positivity of solution

I have a non-negative solution $u \in L^2(0,T;H^1) \cap H^1(0,T;(H^1)')$ of the heat equation $$u_t-\Delta u =0$$ on bounded $C^1$ domain $\Omega$, with the boundary condition $$\frac{\partial u(t,x)} …
ChristopherSail's user avatar