Exponential stability analysis and L2-gain analysis are developed for uncertain distributed parameter systems. Scalar heat processes and distributed mechanical oscillators, governed by semilinear partial differential equations of parabolic and, respectively, hyperbolic types, are chosen for treatment. Sufficient exponential stability conditions with a given decay rate are derived in the form of...

Numerical studies of beam filamentation in laser-produced plasma are presented. This involves the numerical solution of the parabolic wave equation. known as the Schrodinger equation, coupled with the thermal transport equations for both ions and electrons, in two dimensions. The solutionof the resultingequationwith non-linear refractive index due to thermal and pondermotive forces. shows self-...

It is well known that the parabolic partial differential equations in two or more space dimensions with overspecified boundary data, feature in the mathematical modeling of many phenomena. In this article, an inverse problem of determining an unknown time-dependent source term of a parabolic equation in general dimensions is considered. Employing some transformations, we change the inverse prob...

A higher-order parabolic equation is used to compute the traveltime (phase) and the amplitude in constant density acoustic media. This approach is in the frequency domain, thereby avoiding the high frequency approximation inherent in the Eikonal equation. Intrinsic attenuation can be naturally incorporated into the calculation. The error at large angles of propagation caused by the expansion of...

Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory. Abstract We obtain solutions of the nonlinear degenerate parabolic equation ∂ ρ ∂ t = div ρ ∇c ⋆ [ ∇ (F ′ (ρ) + V) ] as a steepest descent of an energy with respect to a convex cost functional. The method used here is variational. It requires less uniform convexity assumption than that imposed by Alt and ...

Numerical simulation of high frequency waves in highly heterogeneous media is a challenging problem. Resolving the fine structure of the wave field typically requires extremely small time steps and spatial meshes. We show that capturing macroscopic quantities of the wave field, such as the wave energy density, is achievable with much coarser discretizations. We obtain such a result using a time...

Consider the semilinear heat equation ∂tu = ∂ xu + λσ(u)ξ on the interval [0 , 1] with Dirichlet zero boundary condition and a nice nonrandom initial function, where the forcing ξ is space-time white noise and λ > 0 denotes the level of the noise. We show that, when the solution is intermittent [that is, when infz |σ(z)/z| > 0], the expected L-energy of the solution grows at least as exp{cλ} an...

Nonlocal problems is a major research area in many branches of modern physics, biotechnology, chemistry and engineering, which arises when it is impossible to determine the boundary values of unknown function and its derivatives. Increasingly often, there arise problems with nonlocal integral boundary conditions, especially in particle diffusion [1] and heat conduction [2, 3]. Partial different...

In this paper we present an alternative viewpoint on recent studies of regularity of solutions to the Navier-Stokes equations in critical spaces. In particular, we prove that mild solutions which remain bounded in the space Ḣ 1 2 do not become singular in finite time, a result which was proved in a more general setting by L. Escauriaza, G. Seregin and V. Šverák using a different approach. We us...