共查询到20条相似文献,搜索用时 15 毫秒
1.
Noriko Mizoguchi 《Journal of Differential Equations》2011,250(1):26-32
A solution u of a Cauchy problem for a semilinear heat equation
2.
We consider a Cauchy problem for a semilinear heat equation with p>pS where pS is the Sobolev exponent. If u(x,t)=(T−t)−1/(p−1)φ((T−t)−1/2x) for xRN and t[0,T), where φ is a regular positive solution of(P) then u is called a backward self-similar blowup solution. It is immediate that (P) has a trivial positive solution κ≡(p−1)−1/(p−1) for all p>1. Let pL be the Lepin exponent. Lepin obtained a radial regular positive solution of (P) except κ for pS<p<pL. We show that there exist no radial regular positive solutions of (P) which are spatially inhomogeneous for p>pL. 相似文献
3.
We study blow-up of radially symmetric solutions of the nonlinear heat equation ut=Δu+|u|p−1u either on RN or on a finite ball under the Dirichlet boundary conditions. We assume and that the initial data is bounded, possibly sign-changing. Our first goal is to establish various characterizations of type I and type II blow-ups. Among many other things we show that the following conditions are equivalent: (a) the blow-up is of type II; (b) the rescaled solution w(y,s) converges to either φ∗(y) or −φ∗(y) as s→∞, where φ∗ denotes the singular stationary solution; (c) u(x,T)/φ∗(x) tends to ±1 as x→0, where T is the blow-up time.Our second goal is to study continuation beyond blow-up. Among other things we show that if a blow-up is of type I and incomplete, then its limit L1 continuation becomes smooth immediately after blow-up, and that type I blow-up implies “type I regularization,” that is, (t−T)1/(p−1)‖u(⋅,t)L∞‖ is bounded as t↘T. We also give various criteria for complete and incomplete blow-ups. 相似文献
4.
Noriko Mizoguchi 《Mathematische Annalen》2007,339(4):839-877
A solution u of a Cauchy problem for a semilinear heat equation
is said to undergo Type II blowup at t = T if lim sup Let be the radially symmetric singular steady state. Suppose that is a radially symmetric function such that and (u
0)
t
change sign at most finitely many times. We determine the exact blowup rate of Type II blowup solution with initial data
u
0 in the case of p > p
L
, where p
L
is the Lepin exponent. 相似文献
5.
We consider the blowup rate of solutions for a semilinear heat equation
6.
Noriko Mizoguchi 《Journal of Differential Equations》2006,231(1):182-194
The present paper is concerned with a Cauchy problem for a semilinear heat equation
(P) 相似文献
7.
We study blow-up of radially symmetric solutions of the nonlinear heat equation ut=Δu+|u|p−1u either on RN or on a finite ball under the Dirichlet boundary conditions. We assume that N?3 and . Our first goal is to analyze a threshold behavior for solutions with initial data u0=λv, where v∈C∩H1 and v?0, v?0. It is known that there exists λ?>0 such that the solution converges to 0 as t→∞ if 0<λ<λ?, while it blows up in finite time if λ?λ?. We show that there exist at most finitely many exceptional values λ1=λ?<λ2<λk such that, for all λ>λ? with λ≠λj (j=1,2,…,k), the blow-up is complete and of type I with a flat local profile. Our method is based on a combination of the zero-number principle and energy estimates. In the second part of the paper, we employ the very same idea to show that the constant solution κ attains the smallest rescaled energy among all non-zero stationary solutions of the rescaled equation. Using this result, we derive a sharp criterion for no blow-up. 相似文献
8.
9.
Flávio Dickstein Filomena Pacella Berardino Sciunzi 《Journal of Evolution Equations》2014,14(3):617-633
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 \({L= -\Delta - p | u_{p,\mathcal{K}} | ^{p-1}}\) . 相似文献
10.
11.
In this paper, we prove that if is a radially symmetric, sign-changing stationary solution of the nonlinear heat equation
in the unit ball of , N ≥ 3, with Dirichlet boundary conditions, then the solution of (NLH) with initial value blows up in finite time if |λ − 1| > 0 is sufficiently small and if α is subcritical and sufficiently close to 4/(N − 2).
F. Dickstein was partially supported by CNPq (Brazil). 相似文献
12.
Jerry Bebernes Alberto Bressan Victor A. Galaktionov 《NoDEA : Nonlinear Differential Equations and Applications》1996,3(3):269-286
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. 相似文献
13.
Thierry Cazenave Flávio Dickstein Fred B. Weissler 《Journal of Differential Equations》2009,246(7):2669-568
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. 相似文献
14.
The present paper is concerned with a Cauchy problem for a semilinear heat equation with u0L(RN). A solution u of (P) is said to blow up at t=T<+ if lim suptT|u(t)|=+ with the supremum norm |·| in RN. 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. 相似文献
15.
Shi-Zhong Du 《Journal of Differential Equations》2019,266(9):5942-5970
Refined structures of blowup for non-collapsing maximal solution to a semilinear parabolic equation with 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 for , the Hausdorff dimension of the blowup set for maximal solution whose energy is non-collapsing is shown to be no greater than , 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. 相似文献
16.
Zhengce Zhang Shuang Chen 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2013,64(3):483-491
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 ${p > 2, p-1 < q < \frac{Np}{(N-p)}_+ -1}$ . 相似文献
17.
I. A. Shishmarev 《Differential Equations》2005,41(1):146-149
18.
Noriko Mizoguchi 《Journal of Functional Analysis》2005,220(1):214-227
The present paper is concerned with a Cauchy problem for a semilinear heat equation
(P) 相似文献
19.
Some exact solutions to a nonlinear heat equation are constructed. An initial-boundary value problem is examined for a nonlinear heat equation. To construct solutions, the problem for a partial differential equation of the second order is reduced to a similar problem for a first order partial differential equation. 相似文献
20.
M. O. Korpusov 《Theoretical and Mathematical Physics》2012,172(3):1173-1176
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. 相似文献