Critical global asymptotics in higher-order semilinear parabolic equations

Critical Global Asymptotics in Higher-order Semilinear Parabolic Equations

We consider a higher-order semilinear parabolic equationut =−(−∆)mu−g(x,u) in RN×R+, m>1. The nonlinear term is homogeneous: g(x,su)≡ |s|P−1sg(x,u) and g(sx,u) ≡ |s|Qg(x,u) for any s ∈ R, with exponents P > 1, and Q > −2m. We also assume that g satisfies necessary coercivity and monotonicity conditions for global existence of solutions with sufficiently small initial data. The equation is invar...

Self-Similar Blow-Up in Higher-Order Semilinear Parabolic Equations

We study the Cauchy problem in R × R+ for one-dimensional 2mth-order, m > 1, semilinear parabolic PDEs of the form (Dx = ∂/∂x) ut = (−1) D x u + |u| u, where p > 1, and ut = (−1) D x u + e u with bounded initial data u0(x). Specifically, we are interested in those solutions that blow up at the origin in a finite time T . We show that, in contrast to the solutions of the classical secondorder pa...

Blow-up Estimates for Higher-order Semilinear Parabolic Equations

Life Span of Blow-up Solutions for Higher-order Semilinear Parabolic Equations

In this article, we study the higher-order semilinear parabolic equation ut + (−∆)u = |u|, (t, x) ∈ R+ × R , u(0, x) = u0(x), x ∈ R . Using the test function method, we derive the blow-up critical exponent. And then based on integral inequalities, we estimate the life span of blow-up solutions.

Orbit Equivalence of Global Attractors of Semilinear Parabolic Differential Equations

We consider global attractors Af of dissipative parabolic equations ut = uxx + f(x, u, ux) on the unit interval 0 ≤ x ≤ 1 with Neumann boundary conditions. A permutation πf is defined by the two orderings of the set of (hyperbolic) equilibrium solutions ut ≡ 0 according to their respective values at the two boundary points x = 0 and x = 1. We prove that two global attractors, Af and Ag, are glo...