Blow‐up of solutions for a viscoelastic wave equation with variable exponents

Abstract

In this paper, we consider a viscoelastic wave equation with variable exponents: utt−Δu+∫0tg(t−s)Δu(s)ds+a|utfalse|m(x)−2ut=b|ufalse|p(x)−2u, where the exponents of nonlinearity p(·) and m(·) are given functions and a,b > 0 are constants. For nonincreasing positive function g, we prove the blow‐up result for the solutions with positive initial energy as well as nonpositive initial energy. We extend the previous blow‐up results to a viscoelastic wave equation with variable exponents.

Nonexistence of global solutions of a delayed wave equation with variable-exponents

“…In this part, our goal is to prove the local existence result for our main problem (1.1) by using Faedo-Galerkin method. We use similar arguments as in [14,16] to get the result. Firstly, we give the lemma which we need:…”

“…In recent years, some other authors investigate hyperbolic type equation with variable exponents (see [8,13,16,18,24]).…”

Blow-up for a viscoelastic von Karman equation with strong damping and variable exponent source terms

“…See [6,7,11] for more properties of a Lebesgue space with variable exponent. By combining the arguments of [10,19], for every (w 0 , w 1 ) ∈ H 2 0 ( ) × L 2 ( ), we can get a unique local solution w of problem (1.1)-(1.4) with w ∈ C(0, T; H 2 0 ( )) ∩ C 1 (0, T; L 2 ( )) and w t ∈ L 2 (0, T; H 1 0 ( )).…”

“…Furthermore, they showed that the solution with negative initial energy blows up in a finite time. Later, Park and Kang [19] improved and complemented the result of [15] by obtaining a blow-up result of solution with certain positive initial energy for a wave equation of memory type. We also refer to a recent work [4] for a nonlinear diffusion system involving variable exponents dependent on spatial and time variables and cross-diffusion terms.…”

General decay and blow up of solutions for a class of inverse problem with elasticity term and variable‐exponent nonlinearities

“…They proved a blowup in finite time with negative initial energy under suitable conditions on g, f and the variable exponent of the ⃗ p(x, t)-Laplace operator. For more results regarding this matter, we refer the reader to previous studies [13][14][15][16][17][18] and the review paper. 19 Motivated by the aforementioned works, in the present paper, we study a class of elastic inverse source problem with variable-exponent nonlinearities.…”