Harnack's inequality of weak type for the parabolic \(p (x)\)-Laplacian (Q2113412)

In this short but very interesting paper, the author establishes Harnack's inequality of weak type for nonnegative bounded supersolutions of the parabolic \(p(x)\)-Laplacian \[ u_t-\operatorname{div} A(x, t, \nabla u)=0\text{ in }Q := D \times [a, b], \] where \(D\) is a bounded Lipschitz domain in \(\mathbb{R}^n\), \(n\geq2\), \(a<b\). The flow \(A\) is a Caratheodory function and satisfies the coercivity and growth conditions \[ A(x, t, \xi)\cdot\xi\geq C_0|\xi|^{p(x)} \] \[ |A(x, t,\xi)|\leq C_1|\xi|^{p(x)-1} \] for all \(\xi\in R^n\) and almost all $(x, t)\in Q$, where \(C_0\) and \(C_1\) are positive constants. The variable exponent \(p\in L^\infty(D)\) is bounded away from 1 and $\infty$ and \(1 < p_1\leq p(x) \leq p_2 <\infty\) for almost all \(x\in D\). The author supposes that the exponent $p(\cdot)$ satisfies the Zhikov's logarithmic condition \[ |p(x)- p(y)|\leq L\frac{1}{ \ln(e + r-1)}\,. \] Under such assumptions, the author proves Harnack's inequality of weak type for nonnegative bounded supersolutions.