Rate estimates of gradient blowup for a heat equation with exponential nonlinearity

We consider a one-dimensional semilinear parabolic equation , for which the spatial derivative of solutions becomes unbounded in finite time while the solutions themselves remain bounded. We establish estimates of blowup rate upper and lower bounds. We prove that in this case the blowup rate does not match the one obtained by the rescaling method.     

Sign-changing stationary solutions and blowup for a nonlinear heat equation in dimension two

Consider the nonlinear heat equation $$v_t -\Delta v=|v|^{p-1}v \qquad \qquad \qquad (NLH)$$ in the unit ball of \({\mathbb{R}^2}\) , with Dirichlet boundary condition. Let \({u_{p,\mathcal{K}}}\) be a radially symmetric, sign-changing stationary solution having a fixed number \({\mathcal{K}}\) of nodal regions. We prove that the solution of (NLH) with initial value \({\lambda u_{p,\mathcal{K}}}\) blows up in finite time if |λ ?1| > 0 is sufficiently small and if p is sufficiently large. The proof is based on the analysis of the asymptotic behavior of \({u_{p,\mathcal{K}}}\) and of the linearized operator \({L= -\Delta - p | u_{p,\mathcal{K}} | ^{p-1}}\) .     

Sign-changing stationary solutions and blowup for the nonlinear heat equation in a ball

In this paper, we prove that if is a radially symmetric, sign-changing stationary solution of the nonlinear heat equation

On Nonexistence of type II blowup for a supercritical nonlinear heat equation

In this paper we study blowup of radially symmetric solutions of the nonlinear heat equation u_t = Δu + |u|^p?1u either on ?^N or on a finite ball under the Dirichlet boundary conditions. We assume that the exponent p is supercritical in the Sobolev sense, that is, We prove that if p_s < p < p^*, then blowup is always of type I, where p^* is a certain (explicitly given) positive number. More precisely, the rate of blowup in the L^∞ norm is always the same as that for the corresponding ODE dv/dt = |v|^p?1v. Because it is known that “type II” blowup (or, equivalently, “fast blowup”) can occur if p > p^*, the above range of exponent p is optimal. We will also derive various fundamental estimates for blowup that hold for any p > p_s and regardless of type of blowup. Among other things we classify local profiles of type I and type II blowups in the rescaled coordinates. We then establish useful estimates for the so‐called incomplete blowup, which reveal that incomplete blowup solutions belong to nice function spaces even after the blowup time. © 2004 Wiley Periodicals, Inc.     

Optimal estimates for blowup rate and behavior for nonlinear heat equations

We first describe all positive bounded solutions of where \input amstex \loadmsbm $(y,s)\in \Bbb R^N\times \Bbb R$ , 1 < p, and (N − 2)p ≤ N + 2. We then obtain for blowup solutions u(t) of uniform estimates at the blowup time and uniform space-time comparison with solutions of u′ = u^p. © 1998 John Wiley & Sons, Inc.     

Global existence and blowup for sign-changing solutions of the nonlinear heat equation

In this paper, given 0<α<2/N, we prove the existence of a function ψ with the following properties. The solution of the equation ut−Δu=α|u|u on RN with the initial condition u(0)=ψ is global. On the other hand, the solution with the initial condition u(0)=λψ blows up in finite time if λ>0 is either sufficiently small or sufficiently large.     

A Liouville property and quasiconvergence for a semilinear heat equation

This paper is concerned with a supercritical semilinear diffusion equation with the power nonlinearity. Via establishing a Liouville-type property, we prove the quasiconvergence (convergence to a set of steady states) of a large class of global solutions. The method of proof relies on similarity variables and invariant manifold ideas.     

Stability of blowup for a parabolic p-Laplace equation with nonlinear source

In this paper, we mainly consider the stability of blowup of solutions for the p-Laplace equation with nonlinear source ${u_t = {div}(|\nabla u|^{p-2}\nabla u) + u^q,\;\;(x,t)\in\mathbb{R}^N \times (0,T)}$ , with the initial value ${u(x,0) = u_0(x) \geq 0}$ , where ${\|u_0 (x)\|_{L^\infty} \leq M}$ and T < ∞ is the blowup time. Under a small oscillation around the radial initial value, we can prove the solution blows up in finite time and obtain the blowup rate estimate of the form ${\|u(\cdot,t)\|_{L^\infty}\leq C(T-t)^{-\frac{1}{q-1}}}$ , where the constant C > 0 is dependent only on N, p, q, and the parameters q and p are expected to be ${p > 2, p-1 < q < \frac{Np}{(N-p)}_+ -1}$ .     

Liouville theorem for Choquard equation with finite Morse indices

In this article, we study the nonexistence of solution with finite Morse index for the following Choquard type equation-△u=∫RN|u(y)|p|x-y|αdy|u(x)|p-2u(x) in RN where N ≥ 3, 0 α min{4, N}. Suppose that 2 p (2 N-α)/(N-2),we will show that this problem does not possess nontrivial solution with finite Morse index. While for p=(2 N-α)/(N-2),if i(u) ∞, then we have ∫_RN∫_RN|u(x)p(u)(y)~p/|x-y|~α dxdy ∞ and ∫_RN|▽u|~2 dx=∫_RN∫_RN|u(x)p(u)(y)~p/|x-y|~αdxdy.     

Global existence and blowup of solutions for a parabolic equation with a gradient term

The author discusses the semilinear parabolic equation with . Under suitable assumptions on and , he proves that, if with , then the solutions are global, while if with 1$">, then the solutions blow up in a finite time, where is a positive solution of , with .

     

Multiple blowup of solutions for a semilinear heat equation

The present paper is concerned with a Cauchy problem for a semilinear heat equation with u₀L(R^N). A solution u of (P) is said to blow up at t=T<+ if lim sup_tT|u(t)|=+ with the supremum norm |·| in R^N. We show that if and N11, then there exists a proper solution u of (P) which blows up at t=T₁, becomes a regular solution for t(T₁,T₂) and blows up again at t=T₂ for some T₁,T₂ with 0<T₁<T₂<+.Mathematics Subject Classification (2000): 35K20, 35K55, 58K57Revised version: 20 July 2004Acknowledgment The author expresses her gratitude to Professor Marek Fila for useful discussion.     

Energy non-collapsing and refined blowup for a semilinear heat equation

Refined structures of blowup for non-collapsing maximal solution to a semilinear parabolic equation

$u_t?△u=|u{|}^{p?1}u$

with $p>1$ are studied. We will prove that the blowup set is empty for non-collapsing blowing-up in subcritical case, and all finite time non-collapsing blowing-up must be refined type II in critical case. When $p>p_S\equiv \frac{N+2}{N?2}$ for $N\ge 3$, the Hausdorff dimension of the blowup set for maximal solution whose energy is non-collapsing is shown to be no greater than $N?2?\frac{4}{p?1}$, which answers a question proposed in [7] positively. At the end of this paper, we also present some new examples of collapsing and non-collapsing blowups.     

Solution blowup for the heat equation with double nonlinearity

We consider a model initial boundary value problem for the heat equation with double nonlinearity. We use a modified Levin method to prove the solution blowup.     

Rate of Type II blowup for a semilinear heat equation

A solution u of a Cauchy problem for a semilinear heat equation

Multiple blowup of solutions for a semilinear heat equation II

The present paper is concerned with a Cauchy problem for a semilinear heat equation

(P)     

A sufficient condition for blowup solutions of nonlinear heat equations

The author discusses the initial-boundary value problem (u_i)_t=Δu_i+f_i(u₁,…,u_m) with and u_i(x,0)=φ_i(x), i=1,…,m, in a bounded domain Ω⊂Rn. Under suitable assumptions on f_i, he proves that, if φ_i?(1+ε₀)ψ_i in , for some small ε₀>0, then the solutions blow up in a finite time, where ψ_i is a positive solution of Δψ_i+f_i(ψ₁,…,ψ_m)?0, with ψ_i|_∂Di=0 for i=1,…,m. If m=1, the initial value can be negative in a subset of Ω.     

On symmetric and nonsymmetric blowup for a weakly quasilinear heat equation

We construct blow-up patterns for the quasilinear heat equation (QHE) $$u_t = \nabla \cdot (k(u)\nabla u) + Q(u)$$ in Ω×(0,T), Ω being a bounded open convex set in ? ^N with smooth boundary, with zero Dirichet boundary condition and nonnegative initial data. The nonlinear coefficients of the equation are assumed to be smooth and positive functions and moreoverk(u) andQ(u)/u ^p with a fixedp>1 are of slow variation asu→∞, so that (QHE) can be treated as a quasilinear perturbation of the well-known semilinear heat equation (SHE) $$u_t = \nabla u) + u^p .$$ We prove that the blow-up patterns for the (QHE) and the (SHE) coincide in a structural sense under the extra assumption $$\smallint ^\infty k(f(e^s ))ds = \infty ,$$ wheref(v) is a monotone solution of the ODEf′(v)=Q(f(v))/v ^p defined for allv?1. If the integral is finite then the (QHE) is shown to admit an infinite number of different blow-up patterns.