Irregular double-phase evolution problem: existence and global regularity

Abstract

We investigate the homogeneous Dirichlet problem for the irregular double-phase evolution equation \[ u_t-\operatorname{div} \left( a(z)|\nabla u|^{p(z)-2} \nabla u + b(z)|\nabla u|^{q(z)-2} \nabla u\right)=f(z),\quad z=(x,t)\in Q_T:=\Omega\times (0,T), \] where $\Omega \subset \mathbb{R}^N$, $N \geq 2$ is a bounded domain, $T>0$, The non-differentiable coefficients $a(z)$, $b(z)$, the free term $f$, and the variable exponents $p$, $q$ are given functions. The coefficients $a$ and $b$ are nonnegative, bounded, satisfy the inequality \[ a(z)+b(z)\geq \alpha \quad \text{in} \ Q_T, \quad \text{and} \quad |\nabla a|, |\nabla b|, a_t, b_t \in L^d(Q_T) \] for some constant $\alpha>0$, and with $d>2$ depending on $\sup p(z)$, $\sup q(z)$, $N$, and the regularity of initial data $u(x,0)$. The free term $f$ and initial data $u(x,0)$ satisfy \[ f\in L^\sigma(Q_T) \ \text{with} \ \sigma>2 \quad \text{and} \quad |\nabla u(x,0)|\in L^{r}(\Omega) \ \text{with} \ r\geq \max \bigg\{2,\sup_{Q_T}p(z),\sup_{Q_T}q(z)\bigg\}. \] The variable exponents $p,q \in C^{0,1}(\overline{Q}_T)$ satisfy the balance condition \[ \frac{2N}{N+2} < p(z), q(z)< +\infty \ \text{in} \ \overline Q_T \quad \text{and} \quad \max\limits_{\overline Q_T}|p(z)-q(z)|< \dfrac{2}{N+2}. \] Under the above assumptions, we establish the existence of a solution, which is obtained as the limit of classical solutions to a family of regularized problems and preserves initial temporal integrability: \[ |\nabla u(\cdot, t)| \in L^r(\Omega) \ \text{for a.e.} \ t \in (0,T), \] gains global higher integrability: \[ |\nabla u|^{\min\{p(z), q(z)\} + s +r} \in L^1(Q_T) \ \text{for any} \ s \in \left(0, \frac{4}{N+2}\right), \] and attains second-order regularity: \[ a(z) |\nabla u|^{\frac{p+r-2}{2}}+b(z) |\nabla u|^{\frac{q+r-2}{2}}\in L^2(0,T;W^{1,2}(\Omega)). \]