Existence, uniqueness, and convergence of solutions of strongly damped wave equations with arithmetic-mean terms

Yıl 2022, Cilt: 5 Sayı: 2, 191 - 212, 30.06.2022

Le Thi Phuong Ngoc , Nguyen Vu Dzung , Nguyen Huu Nhan , Nguyen Thanh Long

https://doi.org/10.53006/rna.1082465

Öz

In this paper, we study the Robin-Dirichlet
problem $(P_{n})$ for a strongly damped wave equation with
arithmetic-mean terms $S_{n}u$ and $\hat{S}_{n}u,$ where
$u$ is the unknown function, $S_{n}u=\tfrac{1}{n} \sum\nolimits_{i=1}^{n}u(\tfrac{i-1}{n},t)$ and $\hat{S}_{n}u= \tfrac{1}{n}\sum\nolimits_{i=1}^{n}u_{x}^{2}(\tfrac{i-1}{n},t)$.
First, under suitable conditions, we prove that, for each $n\in \mathbb{N},$ $(P_{n})$ has a unique weak solution $u^{n}$. Next, we prove that the sequence of solutions $u^{n}$ converge strongly in appropriate spaces to the weak solution $u$ of the problem $(P),$ where $(P)$ is defined by $(P_{n})$ in which the arithmetic-mean terms $S_{n}u$ and $\hat{S} _{n}u$ are replaced by $\int\nolimits_{0}^{1}u(y,t)dy$ and
$\int\nolimits_{0}^{1}u_{x}^{2}(y,t)dy,$ respectively. Finally,
some remarks on a couple of open problems are given.

Anahtar Kelimeler

Robin-Dirichlet problem, Arithmetic-mean terms, Faedo-Galerkin method, Linear recurrent sequence

Kaynakça

• [1] G.F. Carrier, On the nonlinear vibrations problem of elastic string, Quart. J. Appl. Math. 3 (2) (1945) 157-165.
• [2] M.M. Cavalcanti, V.N.D. Cavalcanti, J.S. Prates Filho, J.A. Soriano, Existence and exponential decay for a Kirchhoff-Carrier model with viscosity, J. Math. Anal. Appl. 226 (1) (1998) 40-60.
• [3] M.M. Cavalcanti, V.N.D. Cavalcanti, J.A. Soriano, J.S. Prates Filho, Existence and asymptotic behaviour for a degenerate Kirchhoff-Carrier model with viscosity and nonlinear boundary conditions, Revista Matematica Complutense, 14 (1) (2001) 177-203.
• [4] M.M. Cavalcanti, V.N.D. Cavalcanti, J.A. Soriano, Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation, Adv. Diff. Eqns. 6 (6) (2001) 701-730.
• [5] O.M. Jokhadze, Global Cauchy problem for wave equations with a nonlinear damping term, Differential Equations, 50 (1) (2014) 57-65.
• [6] G.R. Kirchhoff, Vorlesungen über Mathematische Physik: Mechanik, Teuber, Leipzig, 1876, Section 29.7.
• [7] J.L. Lions, Quelques méthodes de résolution des problèmes aux limites nonlinéaires, Dunod; Gauthier-Villars, Paris, 1969.
• [8] P. Massat, Limiting behavior for strongly damped nonlinear wave equations, J. Differ. Eqs. 48 (1983) 334-349.
• [9] N.H. Nhan, L.T.P. Ngoc, N.T. Long, Existence and asymptotic expansion of the weak solution for a wave equation with nonlinear source containing nonlocal term, Bound. Value Prob. (2017) 2017: 87.
• [10] Vittorino Pata, Marco Squassina, On the strongly damped wave equation, Commun. Math. Phys. 253 (2005) 511-533.
• [11] M. Pellicer, J. Solà-Morales, Analysis of a viscoelastic spring-mass model, J. Math. Anal. Appl. 294 (2004) 687-698.
• [12] M. Pellicer, J. Solà-Morales, Spectral analysis and limit behaviours in a spring-mass system, Comm. Pure. Appl. Math. 7 (3) (2008) 563-577.
• [13] R.E. Showalter, Hilbert space methods for partial differential equations, Elec. J. Diff. Eqns. Monograph 01, 1994.
• [14] G. Todorova, E. Vitillaro, Blow-up for nonlinear dissipative wave, J. Math. Anal. Appl. 303 (2005) 242-257.