Self-similar solutions in reaction-diffusion systems
In this paper we examine self-similar solutions to the system , i = 1,…,m, , t > 0, , i = 1,…,m, , where m > 1 and , to describe asymptotics near the blow up point.
Self-similar solutions of convection-diffusion processes.
Self-similarity in chemotaxis systems
We consider a system which describes the scaling limit of several chemotaxis systems. We focus on self-similarity, and review some recent results on forward and backward self-similar solutions to the system.
Semi-analytical approach to initial problems for systems of nonlinear partial differential equations with constant delay
This paper deals with the differential transform method for solving of an initial value problem for a system of two nonlinear functional partial differential equations of parabolic type. We consider non-delayed as well as delayed types of coupling and the different variety of initial functions are thought over. The convergence of solutions and the error estimation to the presented procedure is studied. Two numerical examples for non-delayed and delayed systems are included.
Semi-classical analysis and vanishing properties of solutions to quasilinear equations.
Semilinear parabolic problems on manifolds and applications to the non-compact Yamabe problem.
Semilineare parabolische Differentialgleichungen mit starker Nichtlinearität.
Sets of admissible initial data for porous-medium equations with absorption.
Similarity stabilizes blow-up
The blow-up of solutions to a quasilinear heat equation is studied using a similarity transformation that turns the equation into a nonlocal equation whose steady solutions are stable. This allows energy methods to be used, instead of the comparison principles used previously. Among the questions discussed are the time and location of blow-up of perturbations of the steady blow-up profile.
Single input controllability of a simplified fluid-structure interaction model
In this paper we study a controllability problem for a simplified one dimensional model for the motion of a rigid body in a viscous fluid. The control variable is the velocity of the fluid at one end. One of the novelties brought in with respect to the existing literature consists in the fact that we use a single scalar control. Moreover, we introduce a new methodology, which can be used for other nonlinear parabolic systems, independently of the techniques previously used for the linearized problem....
Single-point blow-up for a semilinear parabolic system
We consider positive solutions of the system ; in a ball or in the whole space, with . Relatively little is known on the blow-up set for semilinear parabolic systems and, up to now, no result was available for this basic system except for the very special case . Here we prove single-point blow-up for a large class of radial decreasing solutions. This in particular solves a problem left open in a paper of A. Friedman and Y. Giga (1987). We also obtain lower pointwise estimates for the final...
Singular integral inequalities and stability of semilinear parabolic equations
Using a method developed by the author for an analysis of singular integral inequalities a stability theorem for semilinear parabolic PDEs is proved.
Singular limit of a transmission problem for the parabolic phase-field model
A transmission problem describing the thermal interchange between two regions occupied by possibly different fluids, which may present phase transitions, is studied in the framework of the Caginalp-Fix phase field model. Dirichlet (or Neumann) and Cauchy conditions are required. A regular solution is obtained by means of approximation techniques for parabolic systems. Then, an asymptotic study of the problem is carried out as the time relaxation parameter for the phase field tends to 0 in one of...
Singularly perturbed periodic parabolic equations with alternating boundary layer type solutions.
Smooth solutions of systems of quasilinear parabolic equations
We consider in this article diagonal parabolic systems arising in the context of stochastic differential games. We address the issue of finding smooth solutions of the system. Such a regularity result is extremely important to derive an optimal feedback proving the existence of a Nash point of a certain class of stochastic differential games. Unlike in the case of scalar equation, smoothness of solutions is not achieved in general. A special structure of the nonlinear hamiltonian seems to be the...
Smooth Solutions of systems of quasilinear parabolic equations
We consider in this article diagonal parabolic systems arising in the context of stochastic differential games. We address the issue of finding smooth solutions of the system. Such a regularity result is extremely important to derive an optimal feedback proving the existence of a Nash point of a certain class of stochastic differential games. Unlike in the case of scalar equation, smoothness of solutions is not achieved in general. A special structure of the nonlinear Hamiltonian seems to be...
Smoothing effect and regularity for linear parabolic equations
Smoothing effect of small analytic solutions to nonlinear Schrödinger equations
Solitary wave and other solutions for nonlinear heat equations
A number of explicit solutions for the heat equation with a polynomial non-linearity and for the Fisher equation is presented. An extended class of non-linear heat equations admitting solitary wave solutions is described. The generalization of the Fisher equation is proposed whose solutions propagate with arbitrary ad hoc fixed velocity.
Solute transport in porous media with equilibrium and non-equilibrium multiple-site adsorpition: Travelling weaves.