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

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 $Q_T$.