Publications (20)16.91 Total impact
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ABSTRACT: We study longterm 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 wellposedness of this problem, global wellposedness when damping dominates the sources (in an appropriate sense) and a blowup 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 blowup result for positive total initial energy.Journal of Mathematical Analysis and Applications 10/2014; 418(2):535–568. · 1.05 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: This is a study of local and global wellposedness 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 blowup result is proved for solutions with negative initial energy.Nonlinear Analysis 08/2014; 105:62–85. · 1.64 Impact Factor 
Article: Systems of nonlinear wave equations with damping and supercritical boundary and interior sources
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ABSTRACT: We consider the local and global wellposedness 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 supercritical 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 01/2014; 366(5). · 1.02 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.Journal of Differential Equations 08/2013; 257(10). · 1.48 Impact Factor  [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 wellknown 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, wellposed in the finiteenergy 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 nondissipative 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 blowup result for weak solutions with nonnegative initial energy.Zeitschrift für angewandte Mathematik und Physik ZAMP 01/2013; 64(3). · 0.94 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 supercritical nonlinear sources on both the interior and the boundary of a bounded domain Ω∈Rn. The presented theorems include Hadamard local wellposedness, global existence, blowup and nonexistence 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. · 0.84 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.Applicable Analysis 01/2012; · 0.71 Impact Factor 
Article: Convex integrals on Sobolev spaces
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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). · 0.63 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, wellposed in the finiteenergy 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 nondissipative 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.Mathematische Nachrichten 08/2011; 284(16):2032  2064. · 0.58 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: This article is concerned with the global wellposedness of the critically and degenerately damped system of nonlinear wave equations in a bounded domain Ω n , n = 1, 2, 3, with Dirichlét 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.Applicable Analysis 08/2010; 89(8):12011227. · 0.71 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We focus on the global wellposedness of the system of nonlinear wave equations utt−Δu+(duk+evl)utm−1ut=f1(u,v)vtt−Δv+(d′vθ+e′uρ)vtr−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.Nonlinear Analysis 01/2010; 72(5):26582683. · 1.64 Impact Factor  [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  m1 u t =F u (u,v)inΩ×(0,∞),v tt Δv+v t  r1 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)=αu+v p+1 +2βuv 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. 01/2009; 2(3):583608.  [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 longtime behavior of solutions.Boletim da Sociedade Paranaense de Matemática. 01/2007;  [Show abstract] [Hide abstract]
ABSTRACT: The article is concerned with the initial boundary value problem to the semilinear equation u tt ▵u+u k j ' (u t )=u p1 u, where p>1 and j ' denotes the derivative of a C 1 convex and real function j . The authors determine the assumptions, under which generalized solutions satisfy variational equality. They exhibit the finite blowup for this class of solutions under suitable interaction between the damping and source parameters.Indiana University Mathematics Journal 01/2007; 56(3):9951022. · 0.42 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: In this article we focus on the global wellposedness of the differential equation u tt −∆u+u k j (u t) = u p−1 u in Ω×(0, T), where j denotes the derivative of a C 1 convex and real valued function j. The interaction between degenerate damping and a source term constitutes the main challenge of the problem. Problems with nondegenerate damping (k = 0) have been studied in the literature (Georgiev and Todorova, 1994; Levine and Serrin, 1997; Vitillaro, 2003). Thus the degeneracy of monotonicity is the main novelty of this work. Depending on the level of interaction between the source and the damping we characterize the domain of the parameters p, m, k, n (see below) for which one obtains existence, regularity or finite time blow up of solutions. More specifically, when p ≤ m+k global existence of generalized solutions in H 1 × L 2 is proved. For p > m + k, solutions blow up in a finite time. Higher energy solutions are studied as well. For H 2 × H 1 initial data we obtain both local and global solutions with the same regularity. Higher energy solutions are also proved to be unique.Control and cybernetics 01/2005; · 0.38 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: In this article we focus on the global wellposedness of the differential equation $u_{tt}  \Delta_{u} + \mid u\mid^k \delta j(u_t) = \mid u\mid^{p1}u in \Omega \times (0,T)$ , where $\delta j$ is a subdifferential 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 01/2005; 357(7):25712611. · 1.02 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We study a nonlinear wave equation on the twodimensional sphere with a blowingup nonlinearity. The existence and uniqueness of a local regular solution are established. Also, the behavior of the solutions is examined. We show that a large class of solutions to the initial value problem quench in finite time.Journal of Mathematical Analysis and Applications 01/2002; · 1.05 Impact Factor 
Article: GLOBAL EXISTENCE AND NONEXISTENCE FOR NONLINEAR WAVE EQUATIONS WITH DAMPING AND SOURCE TERMS
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ABSTRACT: We consider an initialboundary value problem for a nonlinear wave equation in one space dimension. The nonlinearity features the damping term jujm 1ut and a source term of the formjujp 1u ,w ithm; p > 1. We show that whenever m p, then local weak solutions are global. On the other hand, we prove that whenever p>m and the initial energy is negative, then local weak solutions cannot be global, regardless of the size of the initial data.Transactions of the American Mathematical Society 01/2002; 354(9). · 1.02 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: The regularity of solutions to a large class of analytic nonlinear parabolic equations on the twodimensional sphere is considered. In particular, it is shown that these solutions belong to a certain Gevrey class of functions, which is a subset of the set of real analytic functions. As a consequence it can be shown that the Galerkin schemes, based on the spherical harmonics, converge exponentially fast to the exact solutions, as the number of modes involved in the approximation tends to infinity. Furthermore, in the case that the underlying evolution equation has a global attractor, then this global attractor is contained in the space of spatially real analytic functions whose radii of analyticity are bounded uniformly from below.Journal of Dynamics and Differential Equations 01/2000; 12(2):411433. · 0.86 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: In this article we prove a Gevrey class global regularity to the NavierStokes equations on the rotating two dimensional sphere, S 2 a fundamental model that arises naturally in large scale atmospheric dynamics. As a result one concludes the exponential convergence of the spectral Galerkin numerical method, based on spherical harmonic functions. Moreover, we provide an upper bound for the number of asymptotic degrees of freedom for this system. Mathematics Subject Classification (1991). 35Q30, 76D05, 76U05, 58G11, 86Axx. Keywords. Gevrey regularity, Navier−Stokes equations on the sphere, geophysical flows, determining degrees of freedom.Zeitschrift für angewandte Mathematik und Physik ZAMP 01/1999; 50:341360. · 0.94 Impact Factor
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167  Citations  
16.91  Total Impact Points  
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2002–2014

University of Nebraska at Lincoln
 Department of Mathematics
Lincoln, Nebraska, United States
