[Show abstract][Hide abstract] ABSTRACT: This paper investigates a quasilinear wave equation with Kelvin-Voigt damping, utt − Δpu − Δut = f (u), in a bounded domain Ω ⊂ R3 and subject to Dirichlét boundary conditions. The operator Δp, 2 < p < 3, denotes the classical p-Laplacian. The nonlinear term f (u) is a source feedback that is allowed to have a supercritical exponent, in the sense that the associated Nemytskii operator is not locally Lipschitz from W01, p (Ω) into L2(Ω). Under suitable assumptions on the parameters, we prove existence of local weak solutions, which can be extended globally provided the damping term dominates the source in an appropriate sense. Moreover, a blow-up result is proved for solutions with negative initial total energy.
[Show abstract][Hide abstract] ABSTRACT: This note gives a concise summary of results concerning the well-posedness and long-time behavior of (Reissner)–Mindlin–Timoshenko plate equations as presented in Pei et al. (Local and global well-posedness for semilinear Reissner–Mindlin–Timoshenko plate equations, 2013 and Global well-posedness and stability of semilinear Mindlin–Timoshenko system, 2013). The main feature of the considered model is the interplay between nonlinear viscous interior damping and nonlinear source terms. The results include Hadamard local well-posedness, global existence, blow-up theorems, as well as estimates on the uniform energy decay rates.
Proceedings of the 9th ISAAC Congress, DOI: 10.1007/978-3-319-12577-0_36; 03/2015
[Show abstract][Hide abstract] ABSTRACT: We study long-term behavior of Reissner–Mindlin–Timoshenko (RMT) plate systems, focusing on the interplay between nonlinear viscous damping and source terms. The sources may represent restoring forces, but may also be focusing thus potentially amplifying the total energy which is the primary scenario of interest. This work complements [28] which established local well-posedness of this problem, global well-posedness when damping dominates the sources (in an appropriate sense) and a blow-up in the complementary scenario assuming negative “total” initial energy. The current paper develops the potential well theory for the RMT system: it proves global existence for potential well solutions without restricting the source exponents, derives explicit energy decay rates dependent on the order of the damping exponents, and verifies a blow-up result for positive total initial energy.
Journal of Mathematical Analysis and Applications 10/2014; 418(2):535–568. DOI:10.1016/j.jmaa.2014.03.014 · 1.12 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: This paper is concerned with the Cauchy problem for the semilinear wave
equation: $u_{tt}-\Delta u=F(u) \ \mbox{in} \ R^n\times[0, \infty)$, where the
space dimension $n \ge 2$, $F(u)=|u|^p$ or $F(u)=|u|^{p-1}u$ with $p>1$. Here,
the Cauchy data are non-zero and non-compactly supported. Our results on the
blow-up of positive radial solutions (not necessarily radial in low dimensions
$n=2, 3$) generalize and extend the results of Takamura(1995) and Takamura,
Uesaka and Wakasa(2011). The main technical difficulty in the paper lies in
obtaining the lower bounds for the free solution when both initial position and
initial velocity are non-identically zero in even space dimensions.
[Show abstract][Hide abstract] ABSTRACT: This is a study of local and global well-posedness of nonlinearly perturbed Reissner–Mindlin–Timoshenko plate equations. This PDE system represents an extension of the Timoshenko beam model to plates and accounts for shear deformations. The primary feature of the considered model is the interplay between nonlinear viscous interior damping and nonlinear source terms. The main results verify local and global existence of solutions as well as their continuous dependence on the initial data in the appropriate function spaces. Moreover, a blow-up result is proved for solutions with negative initial energy.
[Show abstract][Hide abstract] ABSTRACT: We consider the local and global well-posedness of the coupled nonlinear wave equations u tt -Δu+g 1 (u t )=f 1 (u,v),u tt -Δv+g 2 (v t )=f 2 (u,v) in a bounded domain Ω⊂ℝ n with Robin and Dirichlét boundary conditions on u and v respectively. The nonlinearities f 1 (u,v) and f 2 (u,v) have super-critical exponents representing strong sources, while g 1 (u t ) and g 2 (v t ) act as damping. In addition, the boundary condition also contains a nonlinear source and a damping term. By employing nonlinear semigroups and the theory of monotone operators, we obtain several results on the existence of local and global weak solutions, and uniqueness of weak solutions. Moreover, we prove that such unique solutions depend continuously on the initial data.
Transactions of the American Mathematical Society 05/2014; 366(5). DOI:10.1090/S0002-9947-2014-05772-3 · 1.12 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Presented here is a study of a viscoelastic wave equation with supercritical
source and damping terms. We employ the theory of monotone operators and
nonlinear semigroups, combined with energy methods to establish the existence
of a unique local weak solution. In addition, it is shown that the solution
depends continuously on the initial data and is global provided the damping
dominates the source in an appropriate sense.
[Show abstract][Hide abstract] ABSTRACT: This paper is concerned with a system of nonlinear wave equations with supercritical interior and boundary sources and subject to interior and boundary damping terms. It is well-known that the presence of a nonlinear boundary source causes significant difficulties since the linear Neumann problem for the single wave equation is not, in general, well-posed in the finite-energy space H
1(Ω) × L
2(∂Ω) with boundary data from L
2(∂Ω) (due to the failure of the uniform Lopatinskii condition). Additional challenges stem from the fact that the sources considered in this article are non-dissipative and are not locally Lipschitz from H
1(Ω) into L
2(Ω) or L
2(∂Ω). With some restrictions on the parameters in the system and with careful analysis involving the Nehari Manifold, we obtain global existence of a unique weak solution and establish (depending on the behavior of the dissipation in the system) exponential and algebraic uniform decay rates of energy. Moreover, we prove a blow-up result for weak solutions with nonnegative initial energy.
Zeitschrift für angewandte Mathematik und Physik ZAMP 06/2013; 64(3). DOI:10.1007/s00033-012-0252-6 · 1.11 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: This paper presents a study of the nonlinear wave equation with p-Laplacian damping: u tt -Δu-Δ p u t =f(u) evolving in a bounded domain Ω⊂ℝ n with Dirichlet boundary conditions. The nonlinearity f(u) represents a strong source which is allowed to have a supercritical exponent, i.e., the Nemytski operator f(u) is not locally Lipschitz from H 0 1 (Ω) into L 2 (Ω). The nonlinear term -Δ p u t acts as a strong damping where the -Δ p denotes the p-Laplacian. Under suitable assumptions on the parameters and with careful analysis involving the Nehari manifold, we prove the existence of a global solution and estimate the decay rates of the energy.
Discrete and Continuous Dynamical Systems 12/2012; 32(12). DOI:10.3934/dcds.2012.32.4361 · 0.83 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: This article presents a unified overview of the latest, to date, results on boundary value problems for wave equations with super-critical nonlinear sources on both the interior and the boundary of a bounded domain Ω∈Rn. The presented theorems include Hadamard local wellposedness, global existence, blow-up and non-existence theorems, as well as estimates on the uniform energy dissipation rates for the appropriate classes of solutions.
Mathematics and Computers in Simulation 02/2012; 82(6):1017–1029. DOI:10.1016/j.matcom.2011.04.006 · 0.95 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In this article, we focus on the life span of solutions to the following system of nonlinear wave equations: in a bounded domain Ω ⊂ ℝ with Robin and Dirichlét boundary conditions on u and v, respectively. The nonlinearities f1(u, v) and f2(u, v) represent strong sources of supercritical order, while g1(ut) and g2(vt) represent interior damping. The nonlinear boundary condition on u, namely ∂νu + u + g(ut) = h(u) on Γ, also features h(u), a boundary source, and g(ut), a boundary damping. Under some restrictions on the parameters, we prove that every weak solution to system above blows up in finite time, provided the initial energy is negative.
[Show abstract][Hide abstract] ABSTRACT: We study the global well-posedness of the nonlinear wave equation utt - Δu - Δput = f(u) in a bounded domain ω ⊂ ℝn with Dirichlét boundary conditions. The nonlinearity f(u) represents a strong source which is allowed to have a supercritical exponent; i.e., the Nemytski operator f(u) is not locally Lipschitz from H01(ω) into L2(ω). The nonlinear term -Δput is a strong damping where the -Δp denotes the p-Laplacian (defined below). Un- der suitable assumptions on the parameters and with careful analysis involving the theory of monotone operators, we prove the existence and uniqueness of a local weak solution. Also, such a unique solution de- pends continuously on the initial data from the finite energy space. In addition, we prove that weak solutions are global, provided the exponent of the damping term dominates the exponent of the source.
Advances in Differential Equations 01/2012; 17(1/2). · 1.01 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Let j 0 ,j 1 :ℝ→[0,∞) denote convex functions vanishing at the origin, and let Ω be a bounded domain in ℝ 3 with sufficiently smooth boundary Γ. This paper is devoted to the study of the convex functional J(u)=∫ Ω j 0 (u)dΩ+∫ Γ j 1 (γu)dΓ on the Sobolev space H 1 (Ω). We describe the convex conjugate J * and the subdifferential ∂J. It is shown that the action of ∂J coincides pointwise a.e. in Ω with ∂j 0 (u(x)), and a.e on Γ with ∂j 1 (u(x)). These conclusions are nontrivial because, although they have been known for the subdifferentials of the functionals J 0 (u)=∫ Ω j 0 (u)dΩ and J 1 (u)=∫ Γ j 1 (γu)dΓ, the lack of any growth restrictions on j 0 and j 1 makes the sufficient domain conditio for the sum of two maximal monotone operators ∂J 0 and ∂J 1 infeasible to verify directly. The presented theorems extend the results of H. Brézis [Intégrales convexes dans les espaces de Sobolev. Proc. int. Symp. partial diff. Equ. Geometry normed lin. Spaces I. (French), Isr. J. Math. 13, 9–23, (1972; Zbl 0249.46017)] and fundamentally complement the emerging research literature addressing supercritical damping and sources in hyperbolic PDE’s. These findings rigorously confirm that a combination of supercritical interior and boundary damping feedbacks can be modeled by the subdifferential of a suitable convex functional on the state space.
Journal of Convex Analysis 01/2012; 19(3). · 0.55 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: This article addresses nonlinear wave equations with supercritical interior and boundary sources, and subject to interior and boundary damping. The presence of a nonlinear boundary source alone is known to pose a significant difficulty since the linear Neumann problem for the wave equation is not, in general, well-posed in the finite-energy space H1(Ω) × L2(∂Ω) with boundary data in L2 due to the failure of the uniform Lopatinskii condition. Further challenges stem from the fact that both sources are non-dissipative and are not locally Lipschitz operators from H1(Ω) into L2(Ω), or L2(∂Ω). With some restrictions on the parameters in the model and with careful analysis involving the Nehari Manifold, we obtain global existence of a unique weak solution, and establish exponential and algebraic uniform decay rates of the finite energy (depending on the behavior of the dissipation terms). Moreover, we prove a blow up result for weak solutions with nonnegative initial energy.
[Show abstract][Hide abstract] ABSTRACT: This article is concerned with the global well-posedness of the critically and degenerately damped system of nonlinear wave equations u(tt) - Delta u + (d vertical bar u vertical bar(k) + e vertical bar v vertical bar(l))u(t) = f(1)(u, v) v(tt) - Delta v + (d'vertical bar v vertical bar(theta) + e'vertical bar u vertical bar(rho))v(t) = f(2)(u, v), in a bounded domain Omega subset of R(n), n = 1, 2, 3, with Dirichlet boundary conditions. The nonlinearities f(1)(u, v) and f(2)(u, v) act as a strong source in the system. Under some restriction on the parameters in the system we obtain several results concerning the existence of local solutions, global solutions, uniqueness and the blow up in finite time.
[Show abstract][Hide abstract] ABSTRACT: We focus on the global well-posedness of the system of nonlinear wave equations utt−Δu+(d|u|k+e|v|l)|ut|m−1ut=f1(u,v)vtt−Δv+(d′|v|θ+e′|u|ρ)|vt|r−1vt=f2(u,v), in a bounded domain Ω⊂Rn, n=1,2,3, with Dirichlét boundary conditions. The nonlinearities f1(u,v) and f2(u,v) act as a strong source in the system. Under some restriction on the parameters in the system we obtain several results on the existence of local solutions, global solutions, and uniqueness. In addition, we prove that weak solutions to the system blow up in finite time whenever the initial energy is negative and the exponent of the source term is more dominant than the exponents of both damping terms.
[Show abstract][Hide abstract] ABSTRACT: This paper is concerned with the study of the nonlinearly damped system of wave equations with Dirichlet boundary conditions u tt -Δu+|u t | m-1 u t =F u (u,v)inΩ×(0,∞),v tt -Δv+|v t | r-1 v t =F v (u,v)inΩ×(0,∞), where Ω is a bounded domain in ℝ n , n=1,2,3 with a smooth boundary ∂Ω=Γ and F is a C 1 function given by F(u,v)=α|u+v| p+1 +2β|uv| p+1 2 · Under some conditions on the parameters in the system and with careful analysis involving the Nehari manifold, we obtain several results on the global existence, uniform decay rates, and blow up of solutions of finite time when the initial energy is nonnegative.
Discrete and Continuous Dynamical Systems - Series S 09/2009; 2(3):583-608. DOI:10.3934/dcdss.2009.2.583 · 0.57 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We discuss in this paper some recent development in the study of nonlinear wave equations. In particular, we focus on those results that deal with wave equations that feature two competing forces.One force is a damping term and the other is a strong source. Our central interest here is to analyze the influence of these forces on the long-time behavior of solutions.
Boletim da Sociedade Paranaense de Matematica 06/2007; 25(1-2). DOI:10.5269/bspm.v25i1-2.7427
[Show abstract][Hide abstract] ABSTRACT: In this article we focus on the global well-posedness of the differential equation $u_{tt} - \Delta_{u} + \mid u\mid^k \delta j(u_t) = \mid u\mid^{p-1}u in \Omega \times (0,T)$ , where $\delta j$ is a sub-differential of a continuous convex function j. Under some conditions on j and the parameters in the equations, we obtain several results on the existence of global solutions, uniqueness, nonexistence and propagation of regularity. Under nominal assumptions on the parameters we establish the existence of global generalized solutions. With further restrictions on the parameters we prove the existence and uniqueness of a global weak solution. In addition, we obtain a result on the nonexistence of global weak solutions to the equation whenever the exponent p is greater than the critical value k + m, and the initial energy is negative. We also address the issue of propagation of regularity. Specifically, under some restriction on the parameters, we prove that solutions that correspond to any regular initial data such that $u_0 \in H^2(\Omega) \bigcap H_0^1(\Omega)$ , $u_1 \in H_0^1 (\Omega)$ are indeed strong solutions.
Transactions of the American Mathematical Society 07/2005; 357(7):2571-2611. DOI:10.2307/3845173 · 1.12 Impact Factor