Blow‐up of solution of wave equation with internal and boundary source term and non‐porous viscoelastic acoustic boundary conditions

Abstract

In this paper we study the blow-up of solution of a mixed problem associated to a nonlinear wave equation with dissipative and source term in a bounded domain of ℝ . On a boundary portion of the domain we consider a non-porous viscoelastic acoustic boundary conditions to a non-locally reacting boundary. K E Y W O R D S acoustic boundary conditions, blow-up, non-locally reacting boundary, non-porous boundary, wave equation M S C ( 2 0 1 0 ) 35B44, 35L05, 35L10, 35L20

Well‐posedness and exponential stability for a nonlinear wave equation with acoustic boundary conditions

“…On the other hand, note that the initial conditions of u m and v m are uniformly bounded above, see (35), and M below due to condition (10). These facts, combined with the convergences of the sequences of initial conditions in (18) 3 -( 18) 4 , we deduce that there exists…”

“…The novelty in their results is that the acoustic boundary conditions are imposed with non-locally reacting properties. See also [33,[35][36][37] for more details. Limaco et al [24] considered acoustic wave equations with quadratic nonlinearities when the dimension is restricted to 2 ≤ n < 4.…”

Exponential stabilization of wave equation with acoustic boundary conditions and its application to memory‐type boundary

“…Recently, wave equations with acoustic boundary conditions have been studied by many authors. [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] Compared with previous articles on this subject, the main difference of this article is related with Equations (1) 3 and (2) 3 . In most previous work, for instance, 9,[12][13][14]16,17,19 the memory term of 𝜕 𝜈 u t has not been considered.…”

“…The asymptotic behavior was obtained in Beale 3 (Theorem 2.6). Recently, wave equations with acoustic boundary conditions have been studied by many authors 5–19 …”

Well-posedness and exponential stability for a nonlinear wave equation with acoustic boundary conditions

“…The novelty in their results is that the acoustic boundary conditions are imposed with non-locally reacting properties. See also [42]- [45] for more details.…”