Blow-up of Result in a Nonlinear Wave Equation with Delay and Source Term

In this paper we consider the initial boundary value problem for a nonlinear damping and a delay term of the form: |u_t|^lu_tt- u (x; t) - u_tt+_1|u_t|^m-2u_t\\+_2|u_t(t-)|^m-2u_t(t-)=b|u|^p-2u; with initial conditions and Dirichlet boundary conditions. Under appropriate conditions on; we prove that there are solutions with negative initial energy that blow-up finite time if .

doi:10.5890/DNC.2021.12.012

Authors

In this paper we consider the initial boundary value problem for a nonlinear damping and a delay term of the form: |u_t|^lu_tt- u (x Laboratory of Analysis and Control of Partial Differential Equations, Djillali Liabes University, P. O. Box 89, Sidi Bel Abbes 22000, Algeria University of Mascara, 29000, Algeria University Center of Ain Temouchent, Department of Mathematics, Ain Temouchent 46000, Algeria Author
t) - u_tt+_1|u_t|^m-2u_t\\+_2|u_t(t-)|^m-2u_t(t-)=b|u|^p-2u Laboratory of Analysis and Control of Partial Differential Equations, Djillali Liabes University, P. O. Box 89, Sidi Bel Abbes 22000, Algeria University of Mascara, 29000, Algeria University Center of Ain Temouchent, Department of Mathematics, Ain Temouchent 46000, Algeria Author
with initial conditions and Dirichlet boundary conditions. Under appropriate conditions on Author
we prove that there are solutions with negative initial energy that blow-up finite time if . Author

DOI:

https://doi.org/10.5890/DNC.2021.12.012

Abstract

In this paper we consider the initial boundary value problem for a nonlinear damping and a delay term of the form: $$ |u_t|^{l}u_{tt}-\Delta u (x,t) -\Delta u_{tt}+\mu_1|u_t|^{m-2}u_t\\+\mu_2|u_t(t-\tau)|^{m-2}u_t(t-\tau)=b|u|^{p-2}u, $$ with initial conditions and Dirichlet boundary conditions. Under appropriate conditions on $\mu_1$, $\mu_2$, we prove that there are solutions with negative initial energy that blow-up finite time if $p \geq \max\{l+2,m\}$.

References

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Blow-up of Result in a Nonlinear Wave Equation with Delay and Source Term

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