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Approximation of nonlinear wave equations with nonstandard anisotropic growth conditions

Jonas Haehnle and Andreas Prohl

Abstract

Weak solutions for nonlinear wave equations involving the p(x)-Laplacian, for p : Ω → (1,∞) are constructed as appropriate limits of solutions of an implicit finite element discretization of the problem. A simple fixed-point scheme with appropriate stopping criterion is proposed to conclude convergence for all, discretization, regularization, perturbation, and stopping parameters tending to zero. Computational experiments are included to motivate interesting dynamics, such as blow-up, and asymptotic decay behavior.

Preprint

Problem

problem

Example 5.1

f=-|u|-0.5u, α=0, p(x,y) = 1+2*xy, u0=hat, v0=0
p energy

Example 5.3

f=-|u|4u, α=1, p ≡ 2, u0=10*hat, v0=0
energy

Example 5.5

f=-|u|4u, α=0, p(x,y) = 1+2*xy, u0=5*bricks, v0=0
p energy

Additional Example 1

f=-|u|4u, α=0, p ≡ 2, u0=3*hat, v0=0
energy

Additional Example 2

f=|u|-0.5u, α=0, u0=hat, v0=0
p energy

Additional Example 3

f=|u|-0.5u, α=0, u0=hat, v0=0
p energy

 

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