Initial blow-up rates and universal bounds for nonlinear heat equations

Blow-up in Nonlinear Heat Equations

In this paper we study the blowup problem of nonlinear heat equations. Our result show that for a certain family of initial conditions the solution will blowup in finite time, the blowup parameters satisfy some dynamics which are asymptotic stable, moreover we provide the remainder estimates. Compare to the previous works our approach is analogous to one used in bifurcation theory and our techn...

A 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...

Blow-up at space infinity for nonlinear heat equations

Blow-Up Time for Nonlinear Heat Equations with Transcendental Nonlinearity

where x ∈ R, F is a given nonlinear function and u is unknown. Due to the mathematical and physical importance, existence and uniqueness theories of solutions of nonlinear heat equations have been extensively studied by many mathematicians and physicists, for example, 1–10 and references therein. Unlike other studies, we focus on the nonlinear heat equations with transcendental nonlinearities s...

Pointwise Bounds and Blow-up for Nonlinear Polyharmonic Inequalities

We obtain results for the following question where m ≥ 1 and n ≥ 2 are integers. Question. For which continuous functions f : [0, ∞) → [0, ∞) does there exist a continuous function ϕ : (0, 1) → (0, ∞) such that every C 2m nonnegative solution u(x) of 0 ≤ −∆ m u ≤ f (u) in B 2 (0)\{0} ⊂ R n satisfies u(x) = O(ϕ(|x|)) as x → 0 and what is the optimal such ϕ when one exists? Résumé Nous obtenons d...