WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 21, 2022
Logarithmic Wave Equation Involving Variable-exponent Nonlinearities:well-posedness and Blow-up
Author:
Abstract: In this paper, we focus on a class of existence, uniqueness, and explosion in a nite time of
solving a logarithmic wave equation model with nonlinearities with variable exponents and nonlinear
source terms under homogeneous Dirichlet boundary conditions.
$$u_{tt}-Δu+|u_t|^{m(.)-2}u_t=|u|^{p(.)-2}u \; ln |u|$$
We applied the Faedo-Galerkin method in combination with the Banach xed point theorem to determine
the existence and uniqueness of a local solution in time. Various inequality techniques were used under
appropriate conditions to obtain the blow-up of a solution. This type of equation is related to fluid
dynamics, electrorheological fluids, quantum mechanics theory, nuclear physics, optics, and geophysics.
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Keywords: Wave equations, Logarithmic nonlinearity, variable exponents spaces, Existence, Finite
time blow-up
Pages: 825-837
DOI: 10.37394/23206.2022.21.94