Marek Fila

Boundedness of global solutions of nonlinear diffusion equations

it is well known that there exist choices of u0 for which the corresponding solutions tend to zero as t -+ cc and other choices for which the solutions blow up in finite time. If we are interested in other types of behavior we may consider the possibility (P) u(t, u,,) exists globally

Remarks on blow up for a nonlinear parabolic equation with a gradient term

We consider a nonlinear parabolic equation previously studied by Chipot and Weissler, and Kawohl and Peletier. We give simple sufficient conditions for the presence and absence of L??-blow up.

Blow-up on the boundary: a survey

where m, p > 0 and Ω is either a smoothly bounded domain in R or Ω = R+ = {(x1, x′) : x′ ∈ RN−1, x1 > 0}, ν is the outward normal. Over the past two decades this problem has received considerable interest. For Ω bounded, m = 1 and p > 1 it was shown by Levine and Payne ([LP1]) in 1974 and by Walter ([Wa]) in 1975 that there are solutions which blow up in finite time. This means that lim sup t→T max Ω u(x, t) =∞ for some T <∞.

A note on the quenching rate

We examine the quenching rate near a quenching point of a solution of a semilinear heat equation with singular powerlike absorption. A selfcontained result on similarity profiles allows us to improve a previous quenching theorem by Guo.

IMMEDIATE REGULARIZATION AFTER BLOW-UP

We study solutions of some supercritical parabolic equations which blow up in finite time but continue to exist globally in the weak sense. We show that the minimal continuation becomes regular immediately after the blow up time, and if it blows up again, it can only do so finitely many times.

Existence of global solutions with slow decay and unbounded free boundary for a superlinear Stefan problem

We consider a one-phase Stefan problem for the heat equation with a superlinear reaction term. It is known from a previous work (Ghidouche, Souplet, & Tarzia [5]) that all global solutions are bounded and decay uniformly to 0. Moreover, it was shown in Ghidouche, Souplet, & Tarzia [5] that either: (i) the free boundary converges to a finite limit and the solution decays at an exponential rate, or (ii) the free boundary grows up to infinity and the decay rate is at most polynomial, and it was also proved that small data solutions behave like (i). Here we prove that there exist global solutions with slow decay and unbounded free boundary, i.e. of type (ii). Also, we establish uniform a priori estimates for all global solutions. Moreover, we provide a correction to an error in the proof of decay from Ghidouche, Souplet, & Tarzia [5].

Global solutions of the Laplace equation with a nonlinear dynamical boundary condition

We study the boundedness and a priori bounds of global solutions of the problem Δu=0 in Ω×(0, T), (∂u/∂t) + (∂u/∂ν) = h(u) on ∂Ω×(0, T), where Ω is a bounded domain in ℝN, ν is the outer normal on ∂Ω and h is a superlinear function. As an application of our results we show the existence of sign-changing stationary solutions.

Linear and nonlinear heat equations in L~d~e~l~t~a^q spaces and universal bounds for global solutions

Abstract. We develop a theory of both linear and nonlinear heat equations in the weighted Lebesgue spaces

Chapter 14 – Blow-up in Nonlinear Heat Equations from the Dynamical Systems Point of View

L^q_\delta

Stabilization of solutions of weakly singular quenching problems

, where