Blow-up result for a class of wave $p$-Laplace equation with nonlinear dissipation in $\mathbb{R}^{n}$

B. Belhadji ; A. Beniani ; Kh. Zennir

Vladikavkazskij matematičeskij žurnal, Tome 23 (2021) no. 1, pp. 11-19

Résumé

The Laplace equations has been studied in several stages and has gradually developed over the past decades. Beginning with the well-known standard equation $\Delta u=0$, where it has been well studied in all aspects, many results have been found and improved in an excellent manner. Passing to $p$-Laplace equation $\Delta_p u=0$ with a constant parameter, whether in stationary or evolutionary systems, where it experienced unprecedented development and was studied in almost exhaustively. In this article, we consider initial value problem for nonlinear wave equation containing the $p$-Laplacian operator. We prove that a class of solutions with negative initial energy blows up in finite time if $ p\geq r \geq m $, by using contradiction argument. Additional difficulties due to the constant exponents in $\mathbb{R}^n$ are treated in order to obtain the main conclusion. We used a contradiction argument to obtain a condition on initial data such that the solution extinct at finite time. In the absence of the density function, our system reduces to the nonlinear damped wave equation, it has been extensively studied by many mathematicians in bounded domain.