Asymptotic Self - Similar Global Blow - Upfor a Quasilinear Heat
نویسنده
چکیده
We study the asymptotic behaviour near nite blow-up time t = T of the solutions to the one-dimensional degenerate quasilinear parabolic equation u t = (u u x) x + u in R (0; T); > 0; 1 < < + 1; with bounded, nonnegative, compactly supported initial data. This parameter range corresponds to global blow-up where u(x; t) ! 1 as t ! T ? for any x 2 R. We prove that the rescaled function f(; t) = (T ? t) 1=(?1) u((T ? t) m ; t); m = ? (+ 1) 2(? 1) < 0; converges uniformly as t ! T to the unique compactly supported, symmetric self-similar proole () 0 satisfying a nonlinear ODE. The proof is based on the intersections comparison (the Sturmian argument) with a two-parametric family of self-similar solutions.
منابع مشابه
An Asymptotic and Numerical Description ofSelf - Similar Blow - up in
We study the blow-up behaviour of two reaction-diiusion problems with a quasilinear degenerate diiusion and a superlinear reaction. We show that in each case the blow-up is self-similar, in contrast to the linear diiusion limit of each in which the diiusion is only approximately self-similar. We then investigate the limit of the self-similar behaviour and describe the transition from a stable m...
متن کاملBlow-up Behavior for a Quasilinear Parabolic Equation with Nonlinear Boundary Condition
In this paper, we study the solution of an initial boundary value problem for a quasilinear parabolic equation with a nonlinear boundary condition. We first show that any positive solution blows up in finite time. For a monotone solution, we have either the single blow-up point on the boundary, or blow-up on the whole domain, depending on the parameter range. Then, in the single blow-up point c...
متن کاملGlobal existence, stability results and compact invariant sets for a quasilinear nonlocal wave equation on $mathbb{R}^{N}$
We discuss the asymptotic behaviour of solutions for the nonlocal quasilinear hyperbolic problem of Kirchhoff Type [ u_{tt}-phi (x)||nabla u(t)||^{2}Delta u+delta u_{t}=|u|^{a}u,, x in mathbb{R}^{N} ,,tgeq 0;,]with initial conditions $u(x,0) = u_0 (x)$ and $u_t(x,0) = u_1 (x)$, in the case where $N geq 3, ; delta geq 0$ and $(phi (x))^{-1} =g (x)$ is a positive function lying in $L^{N/2}(mathb...
متن کاملExistence of blow - up solutions for quasilinear elliptic equation with nonlinear gradient term
In this paper, we consider the quasilinear elliptic equation in a smooth bounded domain. By using the method of lower and upper solutions, we study the existence, asymptotic behavior near the boundary and uniqueness of the positive blow-up solutions for quasilinear elliptic equation with nonlinear gradient term.
متن کاملTHREE TYPES OF SELF-SIMILAR BLOW-UP FOR THE FOURTH-ORDER p-LAPLACIAN EQUATION WITH SOURCE: VARIATIONAL AND BRANCHING APPROACHES
Self-similar blow-up behaviour for the fourth-order quasilinear p-Laplacian equation with source, ut = −(|uxx| uxx)xx + |u| u in R × R+, where n > 0, p > 1, is studied. Using variational setting for p = n+1 and branching techniques for p 6= n+1, finite and countable families of blow-up patterns of the self-similar form uS(x, t) = (T − t) − 1 p−1 f(y), where y = x/(T − t) , β = − p−(n+1) 2(n+2)(...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 1997