Higher regularity of solutions of singular parabolic equations with variable nonlinearity

ABSTRACT

We study the global regularity of solutions of the homogeneous Dirichlet problem for the parabolic equation with variable nonlinearity $$u_t-\text{div}\left({|\mathrm{\nabla }u|}^{p(x,t)-2}\mathrm{\nabla }u-\mathbf{f}\right)=0\text{in}{Q}_T=\mathrm{\Omega }\times (0,T),$$

where p(x, t), $\mathbf{f}(x,t)$ are given functions of their arguments, $n\ge 2$ and $\frac{2n}{n+2}<p(x,t)\le 2$. Conditions on the data are found that guarantee the existence of a unique strong solution such that $u_t\in L^2(Q_T)$ and $|\mathrm{\nabla }u|\in L^{\infty }(0,T;L^{p(·)}(\mathrm{\Omega }))$. It is shown that if $\mathit{\partial }\mathrm{\Omega }\in C^{1+\mathit{\beta }}$ with $\mathit{\beta }\in (0,1)$, p and $\mathbf{f}$ are Hölder-continuous in $\overline{\mathrm{\Omega }}\times (0,T]$, $D_i{\mathbf{f}}_j\in L^2(Q_T)$ and $|\mathrm{\nabla }p|\in L^{\infty }(Q_T)$, then for every strong solution $D_{ij}^{2}u\in L^2(\mathrm{\Omega }\times (s,T))$ with any $s\in (0,T)$.

KEYWORDS:

• Nonlinear parabolic equations
• singular equations
• variable nonlinearity
• regularity of solutions

AMS SUBJECT CLASSIFICATIONS:

• 35K55
• 35K67
• 35B65

Notes

No potential conflict of interest was reported by the authors.

To the memory of Prof. Vasily Zhikov.

Additional information

Funding

The work of the first author was supported by the Russian Science Foundation [grant number 15-11-20019]. The second author was supported by MICINN, Spain, [grant number MTM2013-43671-P].