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.

Bidomain models are commonly used for studying and simulating electrophysiological waves in the cardiac tissue. Most of the time, the associated PDEs are solved using explicit finite difference methods on structured grids. We propose an implicit finite element method using unstructured grids for an anisotropic bidomain model. The impact and numerical requirements of unstructured grid methods is investigated using a test case with re-entrant waves.

Bidomain models are commonly used for studying and simulating electrophysiological waves in the cardiac tissue. Most of the time, the associated PDEs are solved using explicit finite difference methods on structured grids. We propose an implicit finite element method using unstructured grids for an anisotropic bidomain model. The impact and numerical requirements of unstructured grid methods is investigated using a test case with re-entrant waves.

We study the asymptotic behavior of a semi-discrete numerical approximation for a pair of heat equations $u_t=\Delta u$, $v_t=\Delta v$ in $\Omega \times (0,T)$; fully coupled by the boundary conditions $\frac{\partial u}{\partial \eta }=u^{p_{11}}{v}^{p_{12}}$, $\frac{\partial v}{\partial \eta }=u^{p_{21}}{v}^{p_{22}}$ on $\partial \Omega \times (0,T)$, where $\Omega$ is a bounded smooth domain in ${ℝ}^d$. We focus in the existence or not of non-simultaneous blow-up for a semi-discrete approximation $(U,V)$. We prove that if $U$ blows up in finite time then $V$ can fail to blow up if and only if $p_{11}\>1$ and $p_{21}\<2(p_{11}-1)$, which is the same condition as the one for non-simultaneous blow-up in the continuous problem. Moreover,...

We study the asymptotic behavior of a semi-discrete numerical approximation for a pair of heat equations ut = Δu, vt = Δv in Ω x (0,T); fully coupled by the boundary conditions $\frac{\partial u}{\partial \eta }=u^{p_{11}}{v}^{p_{12}}$, $\frac{\partial v}{\partial \eta }=u^{p_{21}}{v}^{p_{22}}$ on ∂Ω x (0,T), where Ω is a bounded smooth domain in ${ℝ}^d$. We focus in the existence or not of non-simultaneous blow-up for a semi-discrete approximation (U,V). We prove that if U blows up in finite time then V can fail to blow up if and only if p11 > 1 and p21 < 2(p11 - 1) , which is the same condition as...

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....

We consider positive solutions of the system $u_t-\Delta u=v^p$; $v_t-\Delta v=u^q$ in a ball or in the whole space, with $p,q>1$. 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 $p=q$. 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...

We consider a reaction-diffusion-convection equation on the halfline (0,1) with the zero Dirichlet boundary condition at $x=0$. We find a positive selfsimilar solution $u$ which blows up in a finite time $T$ at $x=0$ while $u(x,T)$ remains bounded for $x>0$.

Using a method developed by the author for an analysis of singular integral inequalities a stability theorem for semilinear parabolic PDEs is proved.

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...

We investigate the structure of travelling waves for a model of a fungal disease propagating over a vineyard. This model is based on a set of ODEs of the SIR-type coupled with two reaction-diffusion equations describing the dispersal of the spores produced by the fungus inside and over the vineyard. An estimate of the biological parameters in the model suggests to use a singular perturbation analysis. It allows us to compute the speed and the profile of the travelling waves. The analytical results...

We establish a singular perturbation property for a class of quasilinear parabolic degenerate equations associated with a mixed Dirichlet-Neumann boundary condition in a bounded domain of ${ℝ}^p$, $1\le p\&lt;+\infty$. In order to prove the $L^1$-convergence of viscous solutions toward the entropy solution of the corresponding first-order hyperbolic problem, we refer to some properties of bounded sequences in $L^{\infty }$ together with a weak formulation of boundary conditions for scalar conservation laws.

In this paper we derive a posteriori error estimates for the heat equation. The time discretization strategy is based on a θ-method and the mesh used for each time-slab is independent of the mesh used for the previous time-slab. The novelty of this paper is an upper bound for the error caused by the coarsening of the mesh used for computing the solution in the previous time-slab. The technique applied for deriving this upper bound is independent of the problem and can be generalized to other time...