Boundary value problems for some quasilinear parabolic equations under minimal conditions on the coefficients

Technical Summary: Boundary Value Problems for Quasilinear Parabolic Equations under Minimal Conditions

Problem Statement
The paper investigates boundary value problems for a class of quasilinear parabolic equations of the form:
$$Mu = u_t - \sum_{i,j=1}^n \frac{\partial}{\partial x_i}(a_{ij}(t, x, u)u_{x_j}) + b(t, x, u, \nabla u) = 0$$
defined on a cylinder $Q = (0, T) \times G$, where $G \subset \mathbb{R}^n$ is a bounded domain. The study addresses two primary boundary condition scenarios:

1. Dirichlet conditions: $u|_{t=0} = u_0$ and $u|_S = g(t, x)$, where $S = (0, T) \times \Gamma$.
2. Robin (oblique) conditions: $u|_{t=0} = u_0$ and $\frac{\partial u}{\partial N} + \psi(t, x, u)|_S = 0$, where $\frac{\partial u}{\partial N}$ is the conormal derivative.

The central objective is to establish existence and uniqueness theorems for solutions within the Sobolev space $W^{1,2}_p(Q)$ under minimal smoothness conditions on the coefficients $a_{ij}$, the lower-order term $b$, and the boundary function $\psi$. This contrasts with existing literature (e.g., monographs [1], [2]) which often require Hölder continuity or stronger regularity assumptions on the coefficients.

Methodology
The proof strategy combines several advanced analytical techniques:

• A Priori Estimates: The authors derive sharp bounds for solutions.
  • Maximum Principle: A maximum principle is established to bound the $L^\infty$ norm of the solution. Unlike previous approaches in Hölder spaces, the authors utilize a different method to handle generalized solutions, particularly for the Robin boundary condition (3).
  • Hölder Continuity: The approach from [1] is adapted to prove that solutions are Hölder continuous, which is a prerequisite for handling the nonlinearity in the coefficients.
  • Frozen Coefficients: To obtain $W^{1,2}_p$ estimates, the method of "frozen coefficients" is employed. The nonlinear problem is locally approximated by linear problems with constant (frozen) coefficients, allowing the use of linear theory estimates.
• Continuation in a Parameter: Existence is proven using the method of continuation in a parameter $\gamma \in [0, 1]$. The problem is deformed from a linear heat equation ($\gamma=0$) to the full quasilinear equation ($\gamma=1$).
• Fixed Point Theory: The solvability of the deformed problem is reduced to finding a fixed point of a compact continuous operator. The compactness relies on the embedding theorems of Sobolev spaces into Hölder spaces.
• Uniqueness Analysis: Uniqueness is demonstrated by analyzing the difference between two potential solutions, utilizing specific growth and Lipschitz-type conditions on the function $b$ and the boundary term $\psi$.

Key Contributions and Results
The paper provides theorems establishing the following:

1. Existence and Uniqueness in $W^{1,2}_p(Q)$:

   • Theorem 7 proves the existence of a unique solution $u \in W^{1,2}_p(Q)$ for both Dirichlet and Robin problems, provided $p > (n+2)/2$.
   • The conditions on the data are minimal:
     • Coefficients $a_{ij}$ are continuous in all variables and satisfy uniform ellipticity.
     • Derivatives $a_{ij, x_i}$ and $a_{ij, u}$ satisfy specific integrability and continuity conditions (e.g., $a_{ij, x_i} \in L^{q_3}(Q; C([-R, R]))$).
     • The function $b$ satisfies Carathéodory conditions and specific growth bounds (quadratic in $\nabla u$).
     • For the Robin condition, $\psi$ and its derivatives satisfy analogous Sobolev-continuity conditions.
   • Consistency conditions at $t=0$ (matching initial and boundary data) are required, particularly for $p > 3$ in the Robin case.

2. A Priori Bounds:

   • Theorems 3 and 4 establish $L^\infty$ bounds for solutions depending on the data norms and the ellipticity constant $\delta_0$.
   • Theorem 5 establishes Hölder continuity ($C^{\alpha_1, 2\alpha_1}$) of the solution, which is crucial for the fixed-point argument.
   • Theorem 6 provides the crucial $W^{1,2}_p$ estimate, showing that the norm of the solution is bounded by the norms of the initial and boundary data, independent of the solution itself.

3. Local Solvability:

   • Theorem 8 addresses local solvability for a broader class of data where global smoothness might not hold initially. It proves that if the initial data $u_0$ and its gradient are bounded, a solution exists locally in time ($Q_{\tau_0}$) even if the coefficients $a_{ij}$ and $b$ are only defined/continuous in a neighborhood of the initial data range.

4. Uniqueness Conditions:

   • Distinct conditions are provided for uniqueness based on the value of $p$:
     • For $p > n+2$, standard Lipschitz conditions on $b$ with respect to $u$ and $\nabla u$ (weighted by $L^q$ functions) suffice.
     • For $(n+2)/2 < p \le n+2$, weaker conditions involving polynomial growth in $\nabla u$ are sufficient, provided the coefficients $g_1, g_2$ belong to specific $L^q$ spaces.

Significance and Claims
The paper claims to expose "sharp conditions" on the data that are "far from optimal" in previous works (specifically referencing the Hölder space requirements of [1] and [2]). By working in Sobolev spaces $W^{1,2}_p(Q)$ with $p > (n+2)/2$, the authors allow for coefficients that are merely continuous (or possess generalized derivatives in specific $L^q$ spaces) rather than requiring $C^k$ or Hölder continuity.

The authors highlight that their approach to maximum estimates allows for the treatment of generalized solutions under Robin boundary conditions, a case where previous methods in [1] and [2] were insufficient. The results are intended to facilitate:

• Solvability analysis for inverse parabolic problems.
• Existence and uniqueness proofs under perturbations.
• The development of constructive approximation methods via iterative techniques.

The work is supported by the Russian Science Foundation and the Government of the Khanty-Mansiysk Autonomous Okrug-YUGRA.