On upper bounds for positive solutions of semilinear equations

Suppose that E is a bounded domain of class C^2,λ in and L is a uniformly elliptic operator in E. The set of all positive solutions of the equation Lu=ψ(u) in E was investigated by a number of authors for various classes of functions ψ. In Dynkin and Kuznetsov (Comm. Pure Appl. Math. 51 (1998) 897) we defined, for every Borel subset Γ of ∂E, two such solutions u_Γ?w_Γ. We also introduced a class of solutions u_ν in 1-1 correspondence with a certain class of σ-finite measures ν on ∂E. With every we associated a pair (Γ,ν) where Γ is a Borel subset of ∂E and . We called this pair the fine boundary trace of u and we denoted in tr(u).Let u⊕v stand for the maximal solution dominated by u+v. We say that u belongs to the class if the condition tr(u)=(Γ,ν) implies that u?w_Γ⊕u_ν and we say that u belongs to if the condition tr(u)=(Γ,ν) implies that u?u_Γ⊕u_ν.It was proved in Dynkin and Kuznetsov (1998) that, under minimal assumptions on L and ψ, the class contains all bounded domains. It follows from results of Mselati (Thése de Doctorat de l'Université Paris 6, 2002; C.R. Acad. Sci. Paris Sér. I 332 (2002); Mem. Amer. Math. Soc. (2003), to appear), that all E of the class C⁴ belong to where Δ is the Laplacian and ψ(u)=u². [Mselati proved that, in his case, u_Γ=w_Γ and therefore the condition tr(u)=(Γ,ν) implies u=u_Γ⊕u_ν=w_Γ⊕u_ν.]By modifying Mselati's arguments, we extend his result to ψ(u)=u^α with 1<α?2 and all bounded domains of class C^2,λ.We start from proving a general localization theorem: under broad assumptions on L, ψ if, for every y∈∂E there exists a domain such that E′⊂E and ∂E∩∂E′ contains a neighborhood of y in ∂E.     

Blow-up of solutions of semilinear parabolic differential equations

For a semilinear parabolic initial boundary value problem we establish criterions on blow-up of the solution in finite time and give bounds for the blow-up time. We treat several applications in both finite and infinite domains. For comparison, sufficient conditions are also given for the existence of global solutions.

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Zusammenfassung Für ein semilineares parabolisches Rand- und Anfangswertproblem stellen wir Kriterien für die Explosion der Lösung in endlicher Zeit auf und geben Schranken für die Explosionszeit an. Einige Anwendungen in beschränkten und unbeschränkten Gebieten werden untersucht, wobei wir als Gegenüberstellung auch hinreichende Bedingungen für die Existenz globaler Lösungen angeben.

     

Capacitary estimates of solutions of semilinear parabolic equations

We prove that any positive solution of ${\partial_tu-\Delta u+u^q=0 (q > 1)}$ in ${\mathbb{R}^N \times (0, \infty)}$ with initial trace (F, 0), where F is a closed subset of ${\mathbb{R}^{N}}$ can be represented, up to two universal multiplicative constants, by a series involving the Bessel capacity ${C_{2/q, q^{\prime}}}$ . As a consequence we prove that there exists a unique positive solution of the equation with such an initial trace. We also characterize the blow-up set of u(x, t) when ${t \downarrow 0}$ , by using the “density” of F expressed in terms of the ${C_{2/q, q^{\prime}}}$ -Bessel capacity.     

Long-time extinction of solutions of some semilinear parabolic equations

We study the long-time behavior of solutions of semilinear parabolic equation of the following type t_∂u−Δu+a₀(x)uq=0 where , d₀>0, 1>q>0, and ω is a positive continuous radial function. We give a Dini-like condition on the function ω by two different methods which implies that any solution of the above equation vanishes in a finite time. The first one is a variant of a local energy method and the second one is derived from semi-classical limits of some Schrödinger operators.     

Capacitary representation of positive solutions of semilinear parabolic equations

We give a global bilateral estimate on the maximal solution ${\overline{u}}_F$ of ${?}_{t}u?\mathrm{\Delta }u+u^q=0$ in ${\mathbb{R}}^N\times (0,\infty )$, $q>1$, $N?1$, which vanishes at $t=0$ on the complement of a closed subset $F?{\mathbb{R}}^N$. This estimate is expressed by a Wiener test involving the Bessel capacity $C_{2/q,q^{\prime }}$. We deduce from this estimate that ${\overline{u}}_F$ is σ-moderate in Dynkin's sense. To cite this article: M. Marcus, L. Véron, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

On properties of solutions of semilinear second-order parabolic equations

We consider semilinear second-order parabolic equations whose principal parts may have either divergence or nondivergence form and whose nonlinear terms satisfy conditions of Bernstein-Dini type. We study the qualitative properties of the classical solutions of nondivergence equations and generalized solutions of equations with divergent principal parts: the behavior of solutions in various unbounded domains and near the boundaries of domains, removability of singularities of solutions, vanishing of solutions in unbounded domains, in particular solutions of compact support and uniqueness and continuous dependence on the boundary conditions for solutions of the exterior initial/boundary problem.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 15, pp. 70–107, 1991. Original article submitted June 18, 1987.     

Generalized functions solutions of monotone and semilinear parabolic equations

We investigate monotone semilinear equations in the setting of generalized functions. We derive existence and uniqueness results even in situations where no solution exists in the sense of distributions; next we show it is consistent with continuous solutions whenever they exist. Furthermore we prove that those transient generalized solutions stabilize toward steady state solutions the way weak solutions do. Lastly we point out an example how these techniques can be applied to different type of nonlinearities.     

Well-posedness of difference schemes for semilinear parabolic equations with weak solutions

The well-posedness of difference schemes approximating initial-boundary value problem for parabolic equations with a nonlinear power-type source is studied. Simple sufficient conditions on the input data are obtained under which the weak solutions of the differential and difference problems are globally stable for all 0 ⩽ t ⩽ +∞. It is shown that, if the condition fails, the solution can blow up (become infinite) in a finite time. A lower bound for the blow-up time is established. In all the cases, the method of energy inequalities is used as based on the application of the Chaplygin comparison theorem, Bihari-type inequalities, and their difference analogues. A numerical experiment is used to illustrate the theoretical results and verify two-sided blow-up time estimates.     

On the numerical computation of blowing‐up solutions for semilinear parabolic equations

Theoretical aspects related to the approximation of the semilinear parabolic equation: $u_t=\Delta u+f(u)$\nopagenumbers\end , with a finite unknown ‘blow‐up’ time T_b have been studied in a previous work. Specifically, for ε a small positive number, we have considered coupled systems of semilinear parabolic equations, with positive solutions and ‘mass control’ property, such that: \def\ve{^\varepsilon}$$u_t\ve=\Delta u\ve+f(u\ve)v\ve\qquad v_t\ve=\Delta v\ve‐\varepsilon f(u\ve)v\ve$$\nopagenumbers\end The solution \def\ve{^\varepsilon}$$\{u\ve,v\ve\}$$\nopagenumbers\end of such systems is known to be global. It is shown that $$\|(u^\varepsilon‐u)(\, .\, ,t)\|_\infty\leq C(M_T)\varepsilon$$\nopagenumbers\end , \def\lt{\char'74}$t\leq T \lt T_b$\nopagenumbers\end where $M_T=\|u(\, .\, ,T)\|_\infty$\nopagenumbers\end and $C(M_T)$\nopagenumbers\end is given by (6). In this paper, we suggest a numerical procedure for approaching the value of the blow‐up time T_b and the blow‐up solution u. For this purpose, we construct a sequence $\{M_\eta\}$\nopagenumbers\end , with $\lim_{\eta\rightarrow 0}M_\eta=\infty$\nopagenumbers\end . Correspondingly, for $\varepsilon\leq1/2C(M_\eta+1)=\eta^\alpha$\nopagenumbers\end and \def\lt{\char'74}$0\lt\alpha\lt\,\!1$\nopagenumbers\end , we associate a specific sequence of times $\{T_\varepsilon\}$\nopagenumbers\end , defined by $\|u^\varepsilon(\, .\, ,T_\varepsilon)\|_\infty=M_\eta$\nopagenumbers\end . In particular, when $\varepsilon=\eta\leq\eta^\alpha$\nopagenumbers\end , the resulting sequence $\{T_\varepsilon\equiv T_\eta\}$\nopagenumbers\end , verifies, $\|(u‐u^\eta)(\, .\, ,t)\|_\infty\leq{1\over2}(\eta)^{1‐\alpha}$\nopagenumbers\end , \def\lt{\char'74}$0\leq t\leq T_\eta\lt T_{\rm b}$\nopagenumbers\end with $\lim_{\eta\rightarrow 0}T_\eta=T_{\rm b}$\nopagenumbers\end . The two special cases of a single‐point blow‐up where $f(u)=\lambda{\rm e}^u$\nopagenumbers\end and $f(u)=u^p$\nopagenumbers\end are then studied, yielding respectively sequences $\{M_\eta\}$\nopagenumbers\end of order $O(\ln|\ln(\eta)|)$\nopagenumbers\end and $O(\{|\ln(\eta)|\}^{1/p‐1})$\nopagenumbers\end . The estimate $|T_\eta‐T_{\rm b}|/T_{\rm b}=O(1/|\ln(\eta)|)$\nopagenumbers\end is proven to be valid in both cases. We conduct numerical simulations that confirm our theoretical results. Copyright © 2001 John Wiley & Sons, Ltd.     

A priori estimates of strong solutions of semilinear parabolic equations

We study an initial boundary value problem for the semilinear parabolic equation