Finally, some remarks on a couple of open problems are given.
📌 3 Maddede Bu Makale
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Amaç & Hipotez: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 $\hat{S}_{n}u= \tfrac{1}{n}\sum\nolimits_{i=1}^{n}u_{x}^{2}(\tfrac{i-1}{n},t)$.
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Temel Bulgular & Veri: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.
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Sonuç & Katkı:Finally, some remarks on a couple of open problems are given.
🔬Tür:Araştırma Makalesi
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:Robin-Dirichlet problem · Arithmetic-mean terms · Faedo-Galerkin method
Abstract
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.
Ngoc, L. T. P., Dzung, N. V., Nhan, N. H., Long, N. T. (2022). Existence, uniqueness, and convergence of solutions of strongly damped wave equations with arithmetic-mean terms. Results in Nonlinear Analysis, Cilt: 5 Sayı: 2, 191–212. https://doi.org/10.53006/rna.1082465
G.F. Carrier, On the nonlinear vibrations problem of elastic string, Quart. J. Appl. Math. 3 (2) (1945) 157-165.
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.
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.
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.
O.M. Jokhadze, Global Cauchy problem for wave equations with a nonlinear damping term, Differential Equations, 50 (1) (2014) 57-65.
J.L. Lions, Quelques méthodes de résolution des problèmes aux limites nonlinéaires, Dunod; Gauthier-Villars, Paris, 1969.
P. Massat, Limiting behavior for strongly damped nonlinear wave equations, J. Differ. Eqs. 48 (1983) 334-349.
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.
Vittorino Pata, Marco Squassina, On the strongly damped wave equation, Commun. Math. Phys. 253 (2005) 511-533.
M. Pellicer, J. Solà-Morales, Analysis of a viscoelastic spring-mass model, J. Math. Anal. Appl. 294 (2004) 687-698.
M. Pellicer, J. Solà-Morales, Spectral analysis and limit behaviours in a spring-mass system, Comm. Pure. Appl. Math. 7 (3) (2008) 563-577.
R.E. Showalter, Hilbert space methods for partial differential equations, Elec. J. Diff. Eqns. Monograph 01, 1994.
G. Todorova, E. Vitillaro, Blow-up for nonlinear dissipative wave, J. Math. Anal. Appl. 303 (2005) 242-257.