Abstract
We study a nonlinear parabolic equation arising from heat combustion and plane curve evolution problems. Suppose that a solution satisfies a symmetry condition and blows up of type Ⅱ. We give an upper bound and a lower bound for the blowup rate of the solution. The lower bound obtained here is probably optimal.
Highlights
The present paper concerns the blowup rate of non-negative solutions of the equation ut = up(uxx + u), x ∈ (−L, L) t ∈ (0, T ), p ≥ 2. (1.1)Our goal is to obtain upper bound and lower bound of the blowup rate if a nonnegative solution of (1.1) really blows up of type II at time T
Under the assumption that solutions are periodic with period 2L, u(−L, t) = u(L, t), ux(−L, t) = ux(L, t) for t ∈ (0, T )
Equation (1.1) with Dirichlet boundary condition arises in the modeling of solar flares in astrophysics and other combustion problems
Summary
Suppose that u satisfies Dirichlet or periodic boundary condition, and the assumptions in Theorem. Suppose that u(x, t) is a solution and satisfies Dirichlet boundary condition (1.2) with L > π/2, or, u(x, t) is a periodic solution (1.3) with L = mπ for certain positive integer m ≥ 2. Let v(x, τ ) be a non-constant positive solution of the rescaled equation (2.7) with periodic boundary condition (2.9). Arguments in the proof of Proposition 2.2 cannot be applied to solutions of (2.7) with Dirichlet boundary condition (2.8), due to the fact that the function A(x, τ ) vanishes on the boundary. Let u(x, t) be a solution of (1.1) with Dirichlet boundary condition as describled in Theorem 1.1.
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