We prove an abstract version of the striking diffusion phenomenon that offers a strong connection between the asymptotic behavior of abstract parabolic and dissipative hyperbolic equations. An important aspect of our approach is that we use in a natural way spectral analysis without involving complicated resolvent estimates. Our proof of the diffusion phenomenon does not use the individual behavior of solutions; instead we show that only their difference matters. We estimate the Hilbert norm of the difference in terms of the Hilbert norm of solutions to the parabolic problems, which allows us to transfer the decay from the parabolic to the hyperbolic problem. The application of these estimates to operators with Markov property combined with a weighted Nash inequality yields explicit and sharp decay rates for hyperbolic problems with variable (x-dependent) coefficients in exterior domains. Our method provides new insight in this area of extensive research which was not well understood until now.
In this paper we show local (and partially global) in time existence for the Westervelt equation with several versions of nonlinear damping. This enables us to prove well-posedness with spatially varying L∞-coefficients, which includes the situation of interface coupling between linear and nonlinear acoustics as well as between linear elasticity and nonlinear acoustics, as relevant, e.g., in high intensity focused ultrasound (HIFU) applications.
Abstract. We establish weighted L 2 −estimates for dissipative wave equations with variable coefficients that exhibit a dissipative term with a space dependent potential. These results yield decay estimates for the energy and the L 2 −norm of solutions. The proof is based on the multiplier method where multipliers are specially engineered from asymptotic profiles of related parabolic equations.
We study the asymptotic behavior of solutions to dissipative wave equations involving two non-commuting self-adjoint operators in a Hilbert space. The main result is that the abstract diffusion phenomenon takes place, as solutions of such equations approach solutions of diffusion equations at large times. When the diffusion semigroup has the Markov property and satisfies a Nash-type inequality, we obtain precise estimates for the consecutive diffusion approximations and remainder. We present several important applications including sharp decay estimates for dissipative hyperbolic equations with variable coefficients on an exterior domain. In the nonlocal case we obtain the first decay estimates for nonlocal wave equations with damping; the decay rates are sharp.
Abstract. We show local existence of solutions to the initial boundary value problem corresponding to a semilinear wave equation with interior damping and source terms. The difficulty in dealing with these two competitive forces comes from the fact that the source term is not a locally Lipschitz function from H 1 (Ω) into L 2 (Ω) as typically assumed in the literature. The strategy behind the proof is based on the physics of the problem, so it does not use the damping present in the equation. The arguments are natural and adaptable to other settings/other PDEs.
We consider a nonlocal operator as a natural generalization to the biharmonic operator that arises in thin plate theory. The operator is built in the nonlocal calculus framework defined in  and connects with the recent theory of peridynamics. This framework enables us to consider non-smooth approximations to fourth-order elliptic boundary value problems. For these systems we introduce nonlocal formulations of the clamped and hinged boundary conditions that are well-defined even for irregular domains. We demonstrate existence and uniqueness of solutions to these nonlocal problems and demonstrate their L 2 -strong convergence to functions in W 1,2 as the nonlocal interaction horizon goes to zero. For regular domains we identify these limits as the weak solutions of the corresponding classical elliptic boundary value problems. As a part of our proof we also establish that the nonlocal Laplacian of a smooth function is Lipschitz continuous.2010 Mathematics Subject Classification. Primary: 45A05, 45P05. Secondary: 35L35, 74K20.
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