DOI: 10.1515/dema-2024-0048 ISSN: 2391-4661

Absence of global solutions to wave equations with structural damping and nonlinear memory

Mokhtar Kirane, Abderrazak Nabti, Lotfi Jlali

Abstract

We prove the nonexistence of global solutions for the following wave equations with structural damping and nonlinear memory source term

u t t + ( Δ ) α 2 u + ( Δ ) β 2 u t = 0 t ( t s ) δ 1 u ( s ) p d s {u}_{tt}+{\left(-\Delta )}^{\tfrac{\alpha }{2}}u+{\left(-\Delta )}^{\tfrac{\beta }{2}}{u}_{t}=\underset{0}{\overset{t}{\int }}{\left(t-s)}^{\delta -1}{| u\left(s)| }^{p}{\rm{d}}s
and
u t t + ( Δ ) α 2 u + ( Δ ) β 2 u t = 0 t ( t s ) δ 1 u s ( s ) p d s , {u}_{tt}+{\left(-\Delta )}^{\tfrac{\alpha }{2}}u+{\left(-\Delta )}^{\tfrac{\beta }{2}}{u}_{t}=\underset{0}{\overset{t}{\int }}{\left(t-s)}^{\delta -1}{| {u}_{s}\left(s)| }^{p}{\rm{d}}s,
posed in
( x , t ) R N × [ 0 , ) \left(x,t)\in {{\mathbb{R}}}^{N}\times \left[0,\infty )
, where
u = u ( x , t ) u=u\left(x,t)
is the real-valued unknown function,
p > 1 p\gt 1
,
α , β ( 0 , 2 ) \alpha ,\beta \in \left(0,2)
,
δ ( 0 , 1 ) \delta \in \left(0,1)
, by using the test function method under suitable sign assumptions on the initial data. Furthermore, we give an upper bound estimate of the life span of solutions.

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