Boundary blow - up solutions in the unit ball : asymptotics , uniqueness and symmetry
نویسنده
چکیده
Abstract We calculate the full asymptotic expansion of boundary blow-up solutions (see equation (1) below), for any nonlinearity f . Our approach enables us to state sharp qualitative results regarding uniqueness and radial symmetry of solutions, as well as a characterization of nonlinearities for which the blow-up rate is universal. Lastly, we study in more detail the standard nonlinearities f(u) = u, p > 1.
منابع مشابه
Back to the Keller-Osserman condition for boundary blow-up solutions
This article is concerned with the existence, uniqueness and numerical approximation of boundary blow up solutions for elliptic PDE’s as ∆u = f(u) where f satisfies the so-called Keller-Osserman condition. We characterize existence of such solutions for non-monotone f . As an example, we construct an infinite family of boundary blow up solutions for the equation ∆u = u(1 + cos u) on a ball. We ...
متن کاملExact Multiplicity for Boundary Blow-up Solutions
The singularly perturbed boundary blow-up problem −ε2∆u = u(u− a)(1− u) u > 0 in B, u = ∞ on ∂B is studied in the unit ball B ⊂ R (N ≥ 2), a ∈ (1/2, 1) is a constant. It is shown that there exist exactly three positive solutions for the problem and all of them are radially symmetric solutions.
متن کاملA note on critical point and blow-up rates for singular and degenerate parabolic equations
In this paper, we consider singular and degenerate parabolic equations$$u_t =(x^alpha u_x)_x +u^m (x_0,t)v^{n} (x_0,t),quadv_t =(x^beta v_x)_x +u^q (x_0,t)v^{p} (x_0,t),$$ in $(0,a)times (0,T)$, subject to nullDirichlet boundary conditions, where $m,n, p,qge 0$, $alpha, betain [0,2)$ and $x_0in (0,a)$. The optimal classification of non-simultaneous and simultaneous blow-up solutions is determin...
متن کاملLong-time Asymptotics of Solutions of the Second Initial-boundary Value Problem for the Damped Boussinesq Equation
For the damped Boussinesq equation utt−2butxx = −αuxxxx+ uxx + β(u)xx, x ∈ (0, π), t > 0; α, b = const > 0, β = const ∈ R, the second initial–boundary value problem is considered with small initial data. Its classical solution is constructed in the form of a series in small parameter present in the initial conditions and the uniqueness of solutions is proved. The long-time asymptotics is obtain...
متن کاملOn the blow-up rate of large solutions for a porous media logistic equation on radial domain
In this paper we establish the exact blow-up rate of the large solutions of a porous media logistic equation. We consider the carrying capacity function with a general decay rate at the boundary instead of the usual cases when it can be approximated by a distant function. Obtaining the accurate blow-up rate allows us to establish the uniqueness result. Our result covers all previous results on ...
متن کامل