Blow-up of Solutions of a Mixed Problem for Systems of Wave Equations with Boundary Dissipation and with an Interior Nonlinear Focusing Source of Variable Growth Order

Abstract

We study a mixed problem with nonlinear dissipative boundary conditions for systems of one-dimensional semilinear wave equations with a focusing nonlinear source that has a variable growth exponent. Theorems on the blow-up of solutions in finite time are proved.

Blow‐up of solutions of wave equation with a nonlinear boundary condition and interior focusing source of variable order of growth

“…Proof of Lemma To prove this lemma, we use some ideas from the proof of Lemma 3 given in Aliev and Shafieva 30 . If $$$ \rho (u)>1 $$$, then it is obvious that $$$$ {\left(\rho (u)\right)}^{\mathrm{\varkappa}}\le \rho (u).$$…”

“…is a Banach space. [29][30][31]33 The Sobolev space with the variable exponents p(•), that is, W 1 p(•) (0, l), is defined as follows:…”

“…Theorem 3 is proved in a similar way as the proof of Theorem 2 from Aliev and Shafieva. 30 For this reason, here, we only give a scheme for the proof of this theorem.…”

“…To prove this lemma, we use some ideas from the proof of Lemma 3 given in Aliev and Shafieva. 30 If 𝜌(u) > 1, then it is obvious that…”

“…Various problems of mechanics and physics are reduced to the study of initial-boundary value problems for wave equations with a nonlinear boundary condition (see, for example, Rauch 18 ). Numerous studies have been carried out in this direction (see, for example, previous works [19][20][21][22][23][24][25][26][27][28][29][30] and the references indicated in these works). In Feng et al and Liu et al, 28,29 the phenomena of blow-up solutions of the mixed problem for the one-dimensional wave equations with a non-stationary boundary condition were investigated.…”