Monica Marras

Bounds for Blow-Up Time in Nonlinear Parabolic Systems Under Various Boundary Conditions

We consider blow-up solutions to parabolic systems, coupled through their nonlinearities under various boundary conditions with nonlinearities depending on the gradient solution. To obtain a lower bound to blow up time t* for the vector solution, Sobolev-type inequalities are introduced to make use of a differential inequality technique. In addition for Dirichlet systems sufficient conditions are introduced to derive an upper bound for t* and to have a criterion for the global existence of the vector solution.

Reaction-diffusion problems under non-local boundary conditions with blow-up solutions

This paper deals with blow-up solutions to a class of reaction-diffusion equations under non-local boundary conditions. We prove that under certain conditions on the data the blow-up will occur at some finite time and when the blow-up does occur, lower and upper bounds are derived.MSC:35K55, 35K60.

On Extinction Phenomena for Parabolic Systems

We investigate the extinction phenomena for some linear combinations of components of the vector-valued solutions to classes of semilinear parabolic systems. The crucial assumption on simultaneous splitting of the matrix-valued elliptic operators and the nonlinear source term allow us to uncouple the systems into a linear part and a scalar nonlinear equation depending on the solutions of the linear part. We propose necessary conditions and sufficient conditions on the existence of the extinction time for the solutions. We recapture as particular case previous results and apply our abstract theorem to a class of 3 × 3 systems appearing as models in chemical engineering.

Rearrangement Optimization Problems with Free Boundary

This article is concerned with three optimization problems. In the first problem, a functional is maximized with respect to a set that is the weak closure of a rearrangement class; that is, a set comprising rearrangements of a prescribed function. Questions regarding existence, uniqueness, symmetry, and local minimizers are addressed. The second problem is of maximization type related to a Poisson boundary value problem. After defining a relevant function, we prove it is differentiable and derive an explicit formula for its derivative. Further, using the co-area formula, we establish a free boundary result. The third problem is the minimization version of the second problem.

Sharp Upper and Lower Solutions in Some Parabolic Problems

ABSTRACT In this paper we construct upper and lower solutions for a class of parabolic initial-boundary value problems in terms of the solution of the St-Venant problem and first eigenvalue problem. These bounds are sharp in the sense that they coincide with the exact solution in particular situations.

Continuous dependence results for parabolic problems under Robin boundary conditions

We investigate continuous dependence on initial data for solutions of a nonlinear parabolic problem, when Robin conditions are prescribed on the boundary ∂ Ω × (t > 0), Ω a bounded R 2 domain.