On the resolution of the heat equation with discontinuous coefficients
Let \(\Omega\) be an open subset of class \(C^2\) in \(\mathbb{R}^n,\) \(T>0,\) \(f\in L^p(Q)\) and \(p \in (1,\infty).\) The aim of the authors is to study existence and uniqueness with optimal regularity of solutions of the heat equation: \[ {{\partial u}\over {\partial t}}(t,x)= a(t,x) \Delta u(t,x)+ f(t,x) \] where \((t,x) \in Q =]0, T[ \times\Omega\); \(u(t,\sigma)=0\), \((t,\sigma)\in ]0, T[ \times\partial \Omega\) and \(u(x,0)=\psi(x)\), \(\psi\) belonging to suitable Sobolev spaces. Let us also suppose \(a \in L^\infty(Q)\) and \[ \exists \alpha >0,\;\beta > 0: \alpha \leq a(t,x) \leq \beta, \qquad \text{a.e. in }Q. \] Existence and uniqueness with optimal regularity for solutions of the above parabolic equation in nondivergence form are obtained in \(L^q (0, T, L^p(\Omega))\), where \(1<p,q<\infty,\) in two cases. The authors use some results of \textit{D. Lamberton} [J. Funct. Anal. 72, 252-262 (1987; Zbl 0621.47039)] in the first case and in the second case some results of \textit{G. Dore} and \textit{A. Venni} [Math. Z. 186, 189-201 (1987; Zbl 0615.47002)].
existence and uniqueness for nondivergence form equations
scientific article; zbMATH DE number 1423167