Philippe Souplet

Blow-up in nonlocal reaction-diffusion equations

We present new blow-up results for reaction-diffusion equations with nonlocal nonlinearities. The nonlocal source terms we consider are of several types, and are relevant to various models in physics and engineering. They may involve an integral of the unknown function, either in space, in time, or both in space and time, or they may depend on localized values of the solution. For each type of problems, we give finite time blow-up results which significantly improve or extend previous results of several authors. In some cases, when the nonlocal source term is in competition with a local dissipative or convective term, optimal conditions on the parameters for finite time blow-up or global existence are obtained. Our proofs rely on comparison techniques and on a variant of the eigenfunction method combined with new properties on systems of differential inequalities. Moreover, a unified local existence theory for general nonlocal semilinear parabolic equations is developed.

Singularity and decay estimates in superlinear problems via Liouville-type theorems, I: Elliptic equations and systems

In this paper, we study some new connections between Liouville-type theorems and local properties of nonnegative solutions to superlinear elliptic problems. Namely, we develop a general method for derivation of universal, pointwise a priori estimates of local solutions from Liouville-type theorems, which provides a simpler and unified treatment for such questions. The method is based on rescaling arguments combined with a key “doubling” property, and it is different from the classical rescaling method of Gidas and Spruck. As an important heuristic consequence of our approach, it turns out that universal boundedness theorems for local solutions and Liouville-type theorems are essentially equivalent. ∗Supported in part by NSF Grant DMS-0400702 †Supported in part by VEGA Grant 1/3021/06

SHARP GRADIENT ESTIMATE AND YAU'S LIOUVILLE THEOREM FOR THE HEAT EQUATION ON NONCOMPACT MANIFOLDS

We derive a sharp, localized version of elliptic type gradient estimates for positive solutions (bounded or not) to the heat equation. These estimates are related to the Cheng–Yau estimate for the Laplace equation and Hamiltons estimate for bounded solutions to the heat equation on compact manifolds. As applications, we generalize Yaus celebrated Liouville theorem for positive harmonic functions to positive ancient (including eternal) solutions of the heat equation, under certain growth conditions. Surprisingly this Liouville theorem for the heat equation does not hold even in

Finite time blow-up for a non-linear parabolic equation with a gradient term and applications

{\mathbb R}^n

Global solutions of inhomogeneous Hamilton-Jacobi equations

without such a condition. We also prove a sharpened long-time gradient estimate for the log of the heat kernel on noncompact manifolds.

On the growth of mass for a viscous Hamilton-Jacobi equation

We give new finite time blow-up results for the non-linear parabolic equations μ t - Δu = u P and u t - Δu + μ|⊇u| q = u P . We first establish an a priori bound in L P+1 for the positive non-decreasing global solutions. As a consequence, we prove in particular that for the second equation on M N , with q = 2p/(p + 1) and small μ > 0, blow-up can occur for any N ≥ 1, p > 1, (N - 2)p < N + 2 and without energy restriction on the initial data. Incidentally, we present a simple model in population dynamics involving this equation.

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

We consider the viscous Hamilton-Jacobi (VHJ) equationut-Δu=|∇u|p+h(x). For the Dirichlet problem withp>2, it is known thatgradient blow-up may occur in finite time (on the boundary). Whereas considerable effort has been devoted to study the large time behavior of solutions of the equationut-Δu=g(x,u), whereamplitude blow-up may occur if for instanceg(x,u)≈up asu→∞ andp>1, relatively little is known in the case of (VHJ). The aim of this paper is to investigate this question. More precisely, we study the relations between(i)the existence of global classical solutions(ii)the existence of stationary solutions (with gradient possibly singular on the boundary);and we obtain a precise description of the global dynamics for (VHJ). Namely, we show that (i) implies (ii) and that in this case, all global solutions converge uniformly to the (unique) stationary solution. In the radial case, we prove that, conversely, (ii) implies (i). Moreover, for certain (smooth) functionsh, we obtain the existence of global classical solutions with gradient blowing up in infinite time. For 1p-2 or for the Cauchy problem, all solutions are global, but we establish similar relations between the existence of bounded or locally bounded solutions and the existence of stationary solutions. Our proofs depend on some new gradient estimates of solutions, local and global in space, obtained by Bernstein type arguments. As another consequence of these estimates we prove a parabolic Liouville-type theorem for solutions ofut\t-Δu=│Δu│p in ℝNx(\t-\t8,0). Various other results are obtained, including universal bounds for global solutions.

REGULAR SELF-SIMILAR SOLUTIONS OF THE NONLINEAR HEAT EQUATION WITH INITIAL DATA ABOVE THE SINGULAR STEADY STATE

AbstractWe investigate the large time behavior of positive solutions with finite mass for the viscous Hamilton-Jacobi equationut = Δu + |Δu|p,t>0,x ∈ ℝN, wherep≥1 andu(0,.)=u0≥0,u0≢0,u0∈L1. DenotingI∞=limt→∞‖u(t)‖1≤∞, we show that the asymptotic behavior of the mass can be classified along three cases as follows:• ifp≤(N+2)/(N+1), thenI∞=∞ for allu0;• if (N+2)/(N+1)<p<2, then bothI∞=∞ andI∞<∞ occur;• ifp≥2, thenI∞<∞ for allu0. We also consider a similar question for the equationut=Δu+up.

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

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].

GEOMETRY OF UNBOUNDED DOMAINS, POINCARE INEQUALITIES AND STABILITY IN SEMILINEAR PARABOLIC EQUATIONS

Abstract We prove the existence of positive regular solutions of the Cauchy problem for the nonlinear heat equation ut=Δu+|u|αu, with initial value μV, for all μ>1 close enough to 1, where V is the singular stationary solution in R N . This result is obtained when N>2 and 2 N−2 ∗ , where α ∗ is the critical power for the intersection properties of V with regular stationary solutions. Moreover, for μ as above, there exist at least two positive regular solutions with initial value μV. These results are optimal since it is known that no such solution exists if α⩾α ∗ .