Existence and global second-order regularity for anisotropic parabolic equations with variable growth

TL;DR

This paper proves existence, uniqueness, and regularity of solutions for anisotropic parabolic equations with variable exponents, extending classical results to more general nonlinear PDEs with variable growth conditions.

Contribution

It establishes global second-order regularity and existence results for anisotropic parabolic equations with Lipschitz continuous variable exponents, under specific growth and regularity conditions.

Findings

Unique solutions exist under specified conditions.

Solutions exhibit higher integrability and regularity.

Results extend classical theory to variable exponent anisotropic equations.

Abstract

We consider the homogeneous Dirichlet problem for the anisotropic parabolic equation \[ u_t-\sum_{i=1}^ND_{x_i}\left(|D_{x_i}u|^{p_i(x,t)-2}D_{x_i}u\right)=f(x,t) \] in the cylinder $\Omega\times (0,T)$, where $\Omega\subset \mathbb{R}^N$, $N\geq 2$, is a parallelepiped. The exponents of nonlinearity $p_i$ are given Lipschitz-continuous functions. It is shown that if $p_i(x,t)>\frac{2N}{N+2}$, \[ \mu=\sup_{Q_T}\dfrac{\max_i p_i(x,t)}{\min_i p_i(x,t)}<1+\dfrac{1}{N}, \quad |D_{x_i}u_0|^{\max\{p_i(\cdot,0),2\}}\in L^1(\Omega),\quad f\in L^2(0,T;W^{1,2}_0(\Omega)), \] then the problem has a unique solution $u\in C([0,T];L^2(\Omega))$ with $|D_{x_i} u|^{p_i}\in L^{\infty}(0,T;L^1(\Omega))$, $u_t\in L^2(Q_T)$. Moreover, \[ |D_{x_i}u|^{p_i+r}\in L^1(Q_T)\quad \text{with some $r=r(\mu,N)>0$},\qquad |D_{x_i}u|^{\frac{p_i-2}{2}}D_{x_i}u\in W^{1,2}(Q_T). \] The assertions remain true for a smooth domain $\Omega$ if $p_i=2$ on the lateral boundary of $Q_T$.