The existence and blow-up of the local solutions for anisotropic parabolic equations

This paper considers the initial-boundary problem of an anisotropic parabolic equation with variable exponents and superlinear source. The existence of a local solution is proved by the parabolically regularized method. While the existence of a global solution is proved, only when the exponents pi(x)=pi are constants, by Levine's concave method. Two kinds of blow-up criteria are established. The core finding of this paper is that the harmonic mean p¯(x) of p→(x)={pi(x)} plays a decisive role in determining the blow-up of the local solution, a new blow-up criterion is discovered. The standard scaling arguments used in isotropic cases become invalid, intrinsic challenges from the anisotropic nature of the equation, the variable exponents pi(x) bring fundamental difficulties in designing compatible test functions and balancing heterogeneous nonlinearities in energy estimates requires novel adaptations of Levine's concave method.