Narrow Results

Due to the considerable complexity involved in the numerical solution of problems of phase change materials, the development of fast parallel algorithms to solve them is very useful. This large complexity can be reduced by developing faster methods for solving, and suitably modifying programming techniques. We consider the phase field model of the system of p.d.e's

The purpose is a uniqueness result for an ill-posed multi-dimensional parabolic system arising in pollution modeling of surface waters like lakes, estuaries or bays. Exploring organic pollution effects mostly needs the recovery of two tracers, the biochemical oxygen demand (BOD) and the dissolved oxygen (DO) densities when only measurements on the dissolved oxygen concentration are available. The particularity of the reaction–diffusion system resides then in the nature of the boundary conditions. Missing boundary data on BOD is compensated by over-determined boundary conditions on DO which induces a strong coupling in the system. We check first the ill-posedness of the problem. Then, a uniqueness theorem is stated. The saddle point theory and tools from the semi-group analysis turn out to be at the basis of the proof. The results established here may serve for the data completion and the identifiability of pointwise pollution sources in multi-dimensional water bodies.

This paper studies the global existence and the asymptotic self-similar behavior of solutions of the semilinear parabolic system ∂_tu=Δu+a₁|u|^p₁-1u+b₁|v|^q₁-1v, ∂_tv=Δv+a₂|v|^p₂-1v+b₂|u|^q₂-1u, on (0,∞)×ℝⁿ, where a₁, b_i∈ℝ and p_i, q_i>1. Let p=min{p₁,p₂, q₁(1+q₂)/(1+q₁), q₂(1+q₁)/(1+q₂)}. Under the condition p>1+2/n we prove the existence of globally decaying mild solutions with small initial data. Some of them are asymptotic, for large time, to self-similar solutions of appropriate asymptotic systems having each one a self-similar structure. All possible asymptotic self-similar behaviors are discussed in terms of exponents p_i, q_i, the space dimension n and the asymptotic spatial profile of the related initial data.

We consider a parabolic system in divergence form with measurable coefficients in a cylindrical space–time domain with nonsmooth base. The associated nonhomogeneous term is assumed to belong to a suitable weighted Orlicz space. Under possibly optimal assumptions on the coefficients and minimal geometric requirements on the boundary of the underlying domain, we generalize the Calderón–Zygmund theorem for such systems by essentially proving that the spatial gradient of the weak solution gains the same weighted Orlicz integrability as the nonhomogeneous term.

We establish optimal Liouville-type theorems for the system of parabolic inequalities

The general finite difference schemes with intrinsic parallelism for the boundary value problem of the semilinear parabolic system of divergence type with bounded measurable coefficients is studied. By the approach of the discrete functional analysis, the existence and uniqueness of the discrete vector solutions of the nonlinear difference system with intrinsic parallelism are proved. Moreover the unconditional stability of the general difference schemes with intrinsic parallelism justified in the sense of the continuous dependence of the discrete vector solution of the difference schemes on the discrete initial data of the original problems in the discrete norms. Finally the convergence of the discrete vector solutions of the certain difference schemes with intrinsic parallelism to the unique generalized solution of the original semilinear parabolic problem is proved.

This paper addresses linear quadratic control optimal problems for non-autonomous linear control systems using strongly continuous quasi semigroups. Riccati equations are implemented to investigate the control optimal problems of cost functional with finite and infinite horizons. The unique optimal pair for the cost functional is determined by the mild solution of the associated closed-loop problem and the feedback control of the solution of the corresponding Riccati equation. In addition for the infinite horizon, the stabilizability of the system is a sufficiency for the solvability to the Riccati equation. An application in a parabolic system is proposed.