Singular Behavior in Nonlinear Parabolic Equations (original) (raw)

Singular solutions of some nonlinear parabolic equations with spatially inhomogeneous absorption

Calculus of Variations and Partial Differential Equations, 2008

We study the limit behaviour of solutions of ∂tu − ∆u + h(|x|) |u| p−1 u = 0 in R N × (0, T) with initial data kδ0 when k → ∞, where h is a positive nondecreasing function and p > 1. If h(r) = r β , β > N (p − 1) − 2, we prove that the limit function u∞ is an explicit very singular solution, while such a solution does not exist if β ≤ N (p−1)−2. If lim infr→0 r 2 ln(1/h(r)) > 0, u∞ has a persistent singularity at (0, t) (t ≥ 0). If r 0 0 r ln(1/h(r)) dr < ∞, u∞ has a pointwise singularity localized at (0, 0).

Well-posedness of initial value problems for singular parabolic equations

Journal of Differential Equations, 2004

We study well-posedness of initial value problems for a class of singular quasilinear parabolic equations in one space dimension. Simple conditions for well-posedness in the space of bounded nonnegative solutions are given, which involve boundedness of solutions of some related linear stationary problems. By a suitable change of unknown, the above results can be applied to classical initial-boundary value problems for parabolic equations with singular coefficients, as the heat equation with inverse square potential. r

Parabolic equations with nonlinear singularities

Nonlinear Analysis: Theory, Methods & …, 2011

We show the existence of positive solutions u ∈ L 2 (0, T ; H 1 0 (Ω)) for nonlinear parabolic problems with singular lower order terms of asymptotetype. More precisely, we shall consider both semilinear problems whose model is

Singular solutions for a convection diffusion equation with absorption

Journal of Mathematical Analysis and Applications, 1992

In this paper we prove the existence of a very singular solution of the Cauchy problem u (x, 0) == 0 if x f:. 0, (a constant) which is more singular at (0,0) thall the fundamental solution of the heat equation if 1 < p < (N + 2)/N and 1~q~(p + 1)/2. We also prove the nonexistence of singular solutions if p~(N +2)/N and 1~q~(p +1)/2. AMS(MOS) subect classifications. 35 B40, 35 1(55. §1 Introduction Consider the CaucllY problell1 Ut-~u-a• \lu q +uP == 0 ill Q == R N X (0, (0) u(x,O) == 0 for x t= o. (1.1) (1.2) where ais a constant vector and a-=I 0. By a solution we mean a nonnegative function u(x,t) which is continuous in Q\ {(O,O)}, and satisfies (1.1) and (1.2) in the classical sense; in particular, u E 02(Q). The behavior of u(x,t) as (x,t)~(0,0), (x,t) E Q is not prescribed so that u may exhibit a singularity at the origin. Nontrivial solution with a singularity at (0,0) can be obtained by considering (1.1) with the initial condition u (x , 0) == cb (x) in R N .

The heat equation with singular nonlinearity and singular initial data

Journal of Differential Equations, 2006

We study the existence, uniqueness and regularity of positive solutions of the parabolic equation u t − u = a(x)u q + b(x)u p in a bounded domain and with Dirichlet's condition on the boundary. We consider here a ∈ L α (Ω), b ∈ L β (Ω) and 0 < q 1 < p. The initial data u(0) = u 0 is considered in the space L r (Ω), r 1. In the main result (0 < q < 1), we assume a, b 0 a.e. in Ω and we assume that u 0 γ d Ω for some γ > 0. We find a unique solution in the space C