Regularity for parabolic equations with time dependent growth (Q1622542)

The paper concentrates on the Hölder continuity of solutions of parabolic equations with time dependent variable nonlinearity exponent. Precisely, the author considers parabolic equations of \(p(t)\)-Laplace type \[ u_t-\text{div }(|\nabla u|^{p(t)-2}\nabla u)=0 \text{ in }\Omega_T:=\Omega\times(0,T]\subset\mathbb{R}^n\times\mathbb{R}, \tag{1} \] where the exponent \(p(\cdot) : (0,T]\to(1,+\infty)\) is a function of the time variable. In the main result of the article, the variable exponent \(p(\cdot)\) is assumed to satisfy two assumptions. The first one is \(\frac{2n}{n+2}<\gamma_1\leq p(\cdot)\leq\gamma_2<+\infty\), for some real constants \(\gamma_1\) and \(\gamma_2\), and the second hypothesis is the so-called \(\log\)-Hölder continuity, that is \[ |p(t_1)-p(t_2)|\leq\frac{c}{-\log|t_1-t_2|},\quad\forall t_1, t_2\in(0,T] \text{with } |t_1-t_2|\leq\frac{1}{2}. \] Then, the author proves that the (spatial) gradient of a weak solutions of (1) is Hölder continuous. The result thus obtained is proved by a direct method that does not use a comparison argument and without additional conditions on the variable exponent. Unlike the case of the constant exponent, the proof is not based on the comparison of the variable exponent with 2. Then one can get the stability of constants in the estimates when \(p(\cdot)\) crosses the value 2. This result is very interesting.