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Regularity and existence of solutions to parabolic equations with nonstandard p(x,t),q(x,t)-growth conditions

El Bahja Hamid

Opuscula Mathematica

2023

Vol. 43, no. 6

759--788

We study the Cauchy–Dirichlet problem for a class of nonlinear parabolic equations driven by nonstandard p(x, t), q(x, t)-growth condition. We prove theorems of existence and uniqueness of weak solutions in suitable Orlicz-Sobolev spaces, derive global and local in time L∞ bounds for the weak solutions.

New efficient time integrators for non-linear parabolic problems

Bujanda B., Jorge J. C.

Opuscula Mathematica

2006

Vol. 26, no. 3

407-419

In this work a new numerical method is constructed for time-integrating multidimensional parabolic semilinear problems in a very efficient way. The method reaches the fourth order in time and it can be combined with standard spatial discretizations of any order to obtain unconditionally convergent numerical algorithms. The main theoretical results which guarantee this property are explained here, as well as the method characteristics which guarantee a very strong reduction of computational cost in comparison with classical discretization methods.