共查询到20条相似文献,搜索用时 31 毫秒
1.
Ting Cheng 《Journal of Differential Equations》2008,244(4):766-802
The blow-up rate estimate for the solution to a semilinear parabolic equation ut=Δu+V(x)|u|p−1u in Ω×(0,T) with 0-Dirichlet boundary condition is obtained. As an application, it is shown that the asymptotic behavior of blow-up time and blow-up set of the problem with nonnegative initial data u(x,0)=Mφ(x) as M goes to infinity, which have been found in [C. Cortazar, M. Elgueta, J.D. Rossi, The blow-up problem for a semilinear parabolic equation with a potential, preprint, arXiv: math.AP/0607055, July 2006], is improved under some reasonable and weaker conditions compared with [C. Cortazar, M. Elgueta, J.D. Rossi, The blow-up problem for a semilinear parabolic equation with a potential, preprint, arXiv: math.AP/0607055, July 2006]. 相似文献
2.
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
3.
Tong Zhang 《Applied mathematics and computation》2010,217(2):801-810
By constructing different auxiliary functions and using Hopf’s maximum principle, the sufficient conditions for the blow-up and global solutions are presented for nonlinear parabolic equation ut = ∇(a(u)b(x)c(t)∇u) + f(x, u, q, t) with different kinds of boundary conditions. The upper bounds of the “blow-up time” and the “upper estimates” of global solutions are provided. Finally, some examples are presented as the application of the obtained results. 相似文献
4.
This paper deals with ut = Δu + um(x, t)epv(0,t), vt = Δv + uq(0, t)env(x,t), subject to homogeneous Dirichlet boundary conditions. The complete classification on non-simultaneous and simultaneous blow-up is obtained by four sufficient and necessary conditions. It is interesting that, in some exponent region, large initial data u0(v0) leads to the blow-up of u(v), and in some betweenness, simultaneous blow-up occurs. For all of the nonnegative exponents, we find that u(v) blows up only at a single point if m > 1(n > 0), while u(v) blows up everywhere for 0 ? m ? 1 (n = 0). Moreover, blow-up rates are considered for both non-simultaneous and simultaneous blow-up solutions. 相似文献
5.
We consider the problem
- u t=u xx+e u whenx ∈ ?,t > 0,
- u(x, 0) =u 0(x) whenx ∈ ?,
6.
Alexander Gladkov Mohammed Guedda 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(13):4573-4580
In this paper we consider a semilinear parabolic equation ut=Δu−c(x,t)up for (x,t)∈Ω×(0,∞) with nonlinear and nonlocal boundary condition u∣∂Ω×(0,∞)=∫Ωk(x,y,t)uldy and nonnegative initial data where p>0 and l>0. We prove some global existence results. Criteria on this problem which determine whether the solutions blow up in finite time for large or for all nontrivial initial data are also given. 相似文献
7.
Zhu Ning 《高校应用数学学报(英文版)》1998,13(3):241-250
ANOTEONTHEBEHAVIOROFBLOW┐UPSOLUTIONSFORONE┐PHASESTEFANPROBLEMSZHUNINGAbstract.Inthispaper,thefolowingone-phaseStefanproblemis... 相似文献
8.
Maisa Khader 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(12):3945-3963
We study the long time behavior of solutions for damped wave equations with absorption. These equations are generally accepted as models of wave propagation in heterogeneous media with space-time dependent friction a(t,x)ut and nonlinear absorption |u|p−1u (Ikawa (2000) [17]). We consider 1<p<(n+2)/(n−2) and separable a(t,x)=λ(x)η(t) with λ(x)∼(1+|x|)−α and η(t)∼(1+t)−β satisfying conditions (A1) or (A2) which are given. The main results are precise decay estimates for the energy, L2 and Lp+1 norms of solutions. We also observe the following behavior: if α∈[0,1), β∈(−1,1) and 0<α+β<1, there are three different regions for the decay of solutions depending on p; if α∈(−∞,0) and β∈(−1,1), there are only two different regions for the decay of the solutions depending on p. 相似文献
9.
Goro Akagi 《Journal of Differential Equations》2007,241(2):359-385
The existence of local (in time) solutions of the initial-boundary value problem for the following degenerate parabolic equation: ut(x,t)−Δpu(x,t)−|u|q−2u(x,t)=f(x,t), (x,t)∈Ω×(0,T), where 2?p<q<+∞, Ω is a bounded domain in RN, is given and Δp denotes the so-called p-Laplacian defined by Δpu:=∇⋅(|∇u|p−2∇u), with initial data u0∈Lr(Ω) is proved under r>N(q−p)/p without imposing any smallness on u0 and f. To this end, the above problem is reduced into the Cauchy problem for an evolution equation governed by the difference of two subdifferential operators in a reflexive Banach space, and the theory of subdifferential operators and potential well method are employed to establish energy estimates. Particularly, Lr-estimates of solutions play a crucial role to construct a time-local solution and reveal the dependence of the time interval [0,T0] in which the problem admits a solution. More precisely, T0 depends only on Lr|u0| and f. 相似文献
10.
Marcelo Montenegro 《Journal of Mathematical Analysis and Applications》2011,384(2):591-596
We prove finite time extinction of the solution of the equation ut−Δu+χ{u>0}(u−β−λf(u))=0 in Ω×(0,∞) with boundary data u(x,t)=0 on ∂Ω×(0,∞) and initial condition u(x,0)=u0(x) in Ω, where Ω⊂RN is a bounded smooth domain, 0<β<1 and λ>0 is a parameter. For every small enough λ>0 there exists a time t0>0 such that the solution is identically equal to zero. 相似文献
11.
L. E. Payne G. A. Philippin S. Vernier Piro 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2010,61(6):999-1007
We consider the blow-up of the solution to a semilinear heat equation with nonlinear boundary condition. We establish conditions on nonlinearities sufficient to guarantee that u(x, t) exists for all time t > 0 as well as conditions on data forcing the solution u(x, t) to blow up at some finite time t*. Moreover, an upper bound for t* is derived. Under somewhat more restrictive conditions, lower bounds for t* are also derived. 相似文献
12.
We establish propagation and spreading properties for nonnegative solutions of nonhomogeneous reaction-diffusion equations of the type:
t∂u−∇⋅(A(t,x)∇u)+q(t,x)⋅∇u=f(t,x,u) 相似文献
13.
In this paper we study the maximal regularity property for non-autonomous evolution equations t∂u(t)+A(t)u(t)=f(t), u(0)=0. If the equation is considered on a Hilbert space H and the operators A(t) are defined by sesquilinear forms a(t,⋅,⋅) we prove the maximal regularity under a Hölder continuity assumption of t→a(t,⋅,⋅). In the non-Hilbert space situation we focus on Schrödinger type operators A(t):=−Δ+m(t,⋅) and prove Lp−Lq estimates for a wide class of time and space dependent potentials m. 相似文献
14.
T.A. Burton 《Nonlinear Analysis: Theory, Methods & Applications》2010,73(12):3873-3882
We consider a scalar integral equation where a∈L2[0,∞), while C(t,s) has a significant singularity, but is convex when t−s>0. We construct a Liapunov functional and show that g(t,x(t))−a(t)∈L2[0,∞) and that x(t)−a(t)→0 pointwise as t→∞. Small perturbations are also added to the kernel. In addition, we consider both infinite and finite delay problems. This paper offers a first step toward treating discontinuous kernels with Liapunov functionals. 相似文献
15.
Jörg Härterich 《Journal of Mathematical Analysis and Applications》2005,307(2):395-414
This paper deals with the singular limit for
L?u:=ut−Fx(u,?ux)−?−1g(u)=0, 相似文献
16.
Xiaojing Yang 《Journal of Differential Equations》2002,183(1):108-131
In this paper, the boundedness of all solutions of the nonlinear equation (?p(x′))′+(p-1)[α?p(x+)−β?p(x−)]+f(x)+g(x)=e(t) is discussed, where e(t)∈C7 is 2πp-periodic, f,g are bounded C6 functions, ?p(u)=∣u∣p−2u, p?2,α,β are positive constants, x+=max{x,0},x−=max{−x,0}. 相似文献
17.
Zhifei Zhang 《Journal of Mathematical Analysis and Applications》2010,363(2):549-558
We discuss the existence of periodic solutions to the wave equation with variable coefficients utt−div(A(x)∇u)+ρ(x,ut)=f(x,t) with Dirichlet boundary condition. Here ρ(x,v) is a function like ρ(x,v)=a(x)g(v) with g′(v)?0 where a(x) is nonnegative, being positive only in a neighborhood of a part of the domain. 相似文献
18.
Alemdar Hasanov Ali Demir Arzu Erdem 《Journal of Mathematical Analysis and Applications》2007,335(2):1434-1451
This article presents a mathematical analysis of input-output mappings in inverse coefficient and source problems for the linear parabolic equation ut=(kx(x)ux)+F(x,t), (x,t)∈ΩT:=(0,1)×(0,T]. The most experimentally feasible boundary measured data, the Neumann output (flux) data f(t):=−k(0)ux(0,t), is used at the boundary x=0. For each inverse problems structure of the input-output mappings is analyzed based on maximum principle and corresponding adjoint problems. Derived integral identities between the solutions of forward problems and corresponding adjoint problems, permit one to prove the monotonicity and invertibility of the input-output mappings. Some numerical applications are presented. 相似文献
19.
We consider the problem of boundary control by displacements at two points x = 0 and x = l of a process described by the Klein-Gordon-Fock equation with a variable coefficient on the finite interval 0 ≤ x ≤ l. For the critical time interval T = l, we obtain a necessary and sufficient condition for the existence of unique boundary functions u(0, t) = µ(t) and u(l, t) = ν(t) bringing the system from an arbitrary initial state at t = 0 into an arbitrary terminal state at t = T. 相似文献
20.
The reaction-diffusion delay differential equation
ut(x,t)−uxx(x,t)=g(x,u(x,t),u(x,t−τ)) 相似文献