Boundedness and Blowup for Nonlinear Degenerate Parabolic Equations
نویسنده
چکیده
The author deals with the quasilinear parabolic equation ut = [uα + g(u)]∆u + buα+1+f(u,∇u) with Dirichlet boundary conditions in a bounded domain Ω, where f and g are lower order terms. He shows that, under suitable conditions on f and g, whether the solution is bounded or blows up in a finite time depends only on the first eigenvalue of −∆ in Ω with Dirichlet boundary condition. For some special cases, the result is sharp.
منابع مشابه
Harnack inequality and continuity of solutions to quasi-linear degenerate parabolic equations with coefficients from Kato-type classes
For a general class of divergence type quasi-linear degenerate parabolic equations with measurable coefficients and lower order terms from non-linear Kato-type classes, we prove local boundedness and continuity of solutions, and the intrinsic Harnack inequality for positive solutions.
متن کاملSpecial Session 42: Global or/and Blowup Solutions for Nonlinear Evolution Equations and Their Applications
This talk discusses global and blowup solutions of the general quasilinear parabolic system ut = ↵(u, v) u + f(u, v,Du) and vt = (u, v) v + g(u, v,Dv) with homogeneous Dirichlet boundary conditions. We will give su cient conditions such that the solutions either exist globally or blow up in a finite time. In special cases, a necessary and su cient condition for global existence is given. We als...
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 کاملBlowup for Degenerate and Singular Parabolic System with Nonlocal Source
We deal with the blowup properties of the solution to the degenerate and singular parabolic system with nonlocal source and homogeneous Dirichlet boundary conditions. The existence of a unique classical nonnegative solution is established and the sufficient conditions for the solution that exists globally or blows up in finite time are obtained. Furthermore, under certain conditions it is prove...
متن کاملLarge time behavior for some nonlinear degenerate parabolic equations
We study the asymptotic behavior of Lipschitz continuous solutions of nonlinear degenerate parabolic equations in the periodic setting. Our results apply to a large class of Hamilton-Jacobi-Bellman equations. Defining Σ as the set where the diffusion vanishes, i.e., where the equation is totally degenerate, we obtain the convergence when the equation is uniformly parabolic outside Σ and, on Σ, ...
متن کامل