Results of exact controllability will be reported for a semilinear thermoelastic system, with control placed in either the mechanical or thermal equation. The controlled partial differential equation (PDE) model will be considered in two cases: (i) rotational forces are present in the PDE, in which case, the underlying dynamics evince hyperbolic behavior; (ii) the rotational inertia parameter i...

This thesis is concerned with the study of the Blow-up phenomena for parabolic problems, which can be defined in a basic way as the inability to continue the solutions up to or after a finite time, the so called blow-up time. Namely, we consider the blow-up location in space and its rate estimates, for special cases of the following types of problems: (i) Dirichlet problems for semilinear equat...

We study blow-up versus global existence of solutions to a model semilinear parabolic equation in metric measure spaces. Applications graphs and Riemannian manifolds are considered, pointing out the occurrence Fujita phenomenon.

In many practical applications of control theory some constraints on the state and/or on the control need to be imposed. In this paper, we prove controllability results for semilinear parabolic equations under positivity constraints on the control, when the time horizon is long enough. As we shall see, in fact, the minimal controllability time turns out to be strictly positive. More precisely, ...

This paper presents a numerical method for verifying the existence and local uniqueness of a solution for an initial-boundary value problem of semilinear parabolic equations. The main theorem of this paper provides a sufficient condition for a unique solution to be enclosed within a neighborhood of a numerical solution. In the formulation used in this paper, the initial-boundary value problem i...

We consider a class of singular perturbations to the stochastic heat equation or semilinear variations thereof. The interesting feature of these perturbations is that, as the small parameter ε tends to zero, their solutions converge to the ‘wrong’ limit, i.e. they do not converge to the solution obtained by simply setting ε = 0. A similar effect is also observed for some (formally) small stocha...

Elliptic equations: 1. Harmonic functions 2. Perron’s method 3. Potential theory 4. Existence results; the method of suband supersolutions 5. Classical maximum principles for elliptic equations 6. More regularity, Schauder’s theory for general elliptic operators 7. The weak solution approach in one space dimension 8. Eigenfunctions for the Sturm-Liouville problem 9. Generalization to more dimen...

Abstract We study a system of parabolic equations consisting double nonlinear equation Forchheimer type coupled with semilinear equation. The describes fluid-like driven for active-passive pedestrian dynamics. structure the nonlinearity coupling allows us to prove uniqueness solutions. provide also stability estimate solutions respect selected parameters.

A semilinear parabolic equation with the Dirichlet boundary condition is examined. The reaction source a mixed nonlinear function. This paper investigates existence and uniqueness of solution. sufficient for solution being blow-up in finite time found.