Quasilinear Cauchy-Dirichlet problem for parabolic equations with $VMO_x$ coefficients

TL;DR

The paper analyzes the strong solvability of the parabolic quasilinear Cauchy-Dirichlet problem $D_tu - a^{ij}(x,t,u)D_{ij}u = f(x,t,u,Du)$ on $Q = \Omega \times (0,T)$ with homogeneous Dirichlet data on the parabolic boundary, allowing principal coefficients to be discontinuous in $x$ in the $VMO_x$ sense while being merely measurable in $t$. Under Carathéodory data, strict parabolicity, and structural conditions including $a^{ij}(\cdot,u) \in VMO_x(Q) \cap L^{\infty}(Q)$ locally in $u$, local continuity in $u$, quadratic growth of $f$ in $Du$, and a sign condition with respect to $u$, the authors prove existence of a strong solution in $W^{2,1}_{n+1}(Q) \cap C(\overline{Q})$ and Hölder continuity of the solution. A key part of the proof combines Krylov-type maximum principles, a priori gradient estimates via the Solonnikov interpolation, and Leray-Schauder fixed-point arguments to handle the nonlinear dependence on $u$. Uniqueness is obtained under additional monotonicity/Lipschitz assumptions on $f$ and $a^{ij}$ being independent of $u$. The results extend linear parabolic theory with partially $VMO$ coefficients to a nonlinear setting and provide a robust framework for problems with discontinuous coefficients, relevant to heat transfer and semiconductor modeling.

Abstract

We study the strong solvability of the Cauchy-Dirichlet problem for parabolic quasilinear equations with discontinuous data. The principal coefficients depend on the point $(x,t)$ and on the solution u, the dependence on x is of VMO type while these are only measurable with respect to t. Assuming suitable structural conditions on the nonlinear terms, we prove existence and uniqueness of the strong solution, which turns out to be also Holder continuous.

Theorems & Definitions (13)

• Definition 2.1
• Theorem 2.2: Existence
• Theorem 2.3: Uniqueness
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Proposition 3.4