共查询到20条相似文献,搜索用时 15 毫秒
1.
In this paper we consider the Cauchy problem of semilinear parabolic equations with nonlinear gradient terms a(x)|u|q−1u|∇u|p. We prove the existence of global solutions and self-similar solutions for small initial data. Moreover, for a class of initial data we show that the global solutions behave asymptotically like self-similar solutions as t→∞. 相似文献
2.
In this paper we study the large time behavior of non-negative solutions to the Cauchy problem of ut=Δum−uq in RN×(0,∞), where m>1 and q=qc≡m+2/N is a critical exponent. For non-negative initial value u(x,0)=u0(x)∈L1(RN), we show that the solution converges, if u0(x)(1+|x|)k is bounded for some k>N, to a unique fundamental solution of ut=Δum, independent of the initial value, with additional logarithmic anomalous decay exponent in time as t→∞. 相似文献
3.
We investigate the large-time behavior of classical solutions to the thin-film type equation ut=−x(uuxxx). It was shown in previous work of Carrillo and Toscani that for non-negative initial data u0 that belongs to H1(R) and also has a finite mass and second moment, the strong solutions relax in the L1(R) norm at an explicit rate to the unique self-similar source type solution with the same mass. The equation itself is gradient flow for an energy functional that controls the H1(R) norm, and so it is natural to expect that one should also have convergence in this norm. Carrillo and Toscani raised this question, but their methods, using a different Lyapunov functions that arises in the theory of the porous medium equation, do not directly address this since their Lyapunov functional does not involve derivatives of u. Here we show that the solutions do indeed converge in the H1(R) norm at an explicit, but slow, rate. The key to establishing this convergence is an asymptotic equipartition of the excess energy. Roughly speaking, the energy functional whose dissipation drives the evolution through gradient flow consists of two parts: one involving derivatives of u, and one that does not. We show that these must decay at related rates—due to the asymptotic equipartition—and then use the results of Carrillo and Toscani to control the rate for the part that does not depend on derivatives. From this, one gets a rate on the dissipation for all of the excess energy. 相似文献
4.
We study the following complex Ginzburg-Landau equation with cubic nonlinearity on for under initial and Dirichlet boundary conditions u(x,0)=h(x) for x∈Ω, u(x,t)=Q(x,t) on ∂Ω where h,Q are given smooth functions. Under suitable conditions, we prove the existence of a global solution in H1. Further, this solution approaches to the solution of the NLS limit under identical initial and boundary data as a,b→0+. 相似文献
5.
We consider a class of Stefan-type problems having a convection term and a pseudomonotone nonlinear diffusion operator. Assuming data in L1, we prove existence, uniqueness and stability in the framework of renormalized solutions. Existence is established from compactness and monotonicity arguments which yield stability of solutions with respect to L1 convergence of the data. Uniqueness is proved through a classical L1-contraction principle, obtained by a refinement of the doubling variable technique which allows us to extend previous results to a more general class of nonlinear possibly degenerate operators. 相似文献
6.
We consider, for p∈(1,2) and q>1, self-similar singular solutions of the equation vt=div(|∇v|p−2∇v)−vq in Rn×(0,∞); here by self-similar we mean that v takes the form v(x,t)=t−αw(|x|t−αβ) for α=1/(q−1) and β=(q+1−p)/p, whereas singular means that v is non-negative, non-trivial, and for all x≠0. That is, we consider the ODE problem
(0.1) 相似文献
7.
Flávio Dickstein 《Journal of Differential Equations》2006,223(2):303-328
We study the Cauchy problem for the nonlinear heat equation ut-?u=|u|p-1u in RN. The initial data is of the form u0=λ?, where ?∈C0(RN) is fixed and λ>0. We first take 1<p<pf, where pf is the Fujita critical exponent, and ?∈C0(RN)∩L1(RN) with nonzero mean. We show that u(t) blows up for λ small, extending the H. Fujita blowup result for sign-changing solutions. Next, we consider 1<p<ps, where ps is the Sobolev critical exponent, and ?(x) decaying as |x|-σ at infinity, where p<1+2/σ. We also prove that u(t) blows up when λ is small, extending a result of T. Lee and W. Ni. For both cases, the solution enjoys some stable blowup properties. For example, there is single point blowup even if ? is not radial. 相似文献
8.
We consider an Allen-Cahn type equation of the form ut=Δu+ε−2fε(x,t,u), where ε is a small parameter and fε(x,t,u)=f(u)−εgε(x,t,u) a bistable nonlinearity associated with a double-well potential whose well-depths can be slightly unbalanced. Given a rather general initial data u0 that is independent of ε, we perform a rigorous analysis of both the generation and the motion of interface. More precisely we show that the solution develops a steep transition layer within the time scale of order ε2|lnε|, and that the layer obeys the law of motion that coincides with the formal asymptotic limit within an error margin of order ε. This is an optimal estimate that has not been known before for solutions with general initial data, even in the case where gε≡0.Next we consider systems of reaction-diffusion equations of the form
9.
W.Y. Chan 《Journal of Computational and Applied Mathematics》2011,235(13):3831-3840
For the problem given by uτ=(ξrumuξ)ξ/ξr+f(u) for 0<ξ<a, 0<τ<Λ≤∞, u(ξ,0)=u0(ξ) for 0≤ξ≤a, and u(0,τ)=0=u(a,τ) for 0<τ<Λ, where a and m are positive constants, r is a constant less than 1, f(u) is a positive function such that limu→c−f(u)=∞ for some positive constant c, and u0(ξ) is a given function satisfying u0(0)=0=u0(a), this paper studies quenching of the solution u. 相似文献
10.
Thomas Elsken 《Journal of Differential Equations》2004,206(1):94-126
For a bounded smooth domain Ω⊂RNx+Ny let Ω?, 0<?, be a family of domains squeezed in y∈RNy direction. On Ω? we consider a reaction-diffusion equation with nonsymmetrical linear part. We show that under natural conditions on the nonlinearity the generated semi-flows have global attractors which in a certain sense have limits, as ?↓0. 相似文献
11.
In this paper, we establish the critical global existence exponent and the critical Fujita exponent for the nonlinear diffusion equation ut=(logσ(1+u)ux)x, in R+×(0,+∞), subject to a logarithmic boundary flux , furthermore give the blow-up rate for the nonglobal solutions. 相似文献
12.
Cyril Imbert 《Journal of Differential Equations》2005,211(1):218-246
In this paper, we investigate the regularizing effect of a non-local operator on first-order Hamilton-Jacobi equations. We prove that there exists a unique solution that is C2 in space and C1 in time. In order to do so, we combine viscosity solution techniques and Green's function techniques. Viscosity solution theory provides the existence of a W1,∞ solution as well as uniqueness and stability results. A Duhamel's integral representation of the equation involving the Green's function permits to prove further regularity. We also state the existence of C∞ solutions (in space and time) under suitable assumptions on the Hamiltonian. We finally give an error estimate in L∞ norm between the viscosity solution of the pure Hamilton-Jacobi equation and the solution of the integro-differential equation with a vanishing non-local part. 相似文献
13.
For a class of general quasilinear equations on S1, we show that, by a very simple maximum principle technique, as long as the solution stays finite, all of its derivatives also remain finite. Some specific examples are given. Under suitable assumptions, we also derive exponential decay of the derivatives of the solution. 相似文献
14.
In this paper, we investigate the spatial dynamics of a nonlocal and time-delayed reaction-diffusion system, which is motivated by an age-structured population model with distributed maturation delay. The spreading speed c*, the existence of traveling waves with the wave speed c?c*, and the nonexistence of traveling waves with c<c* are obtained. It turns out that the spreading speed coincides with the minimal wave speed for monotone traveling waves. 相似文献
15.
Mauricio Bogoya 《Nonlinear Analysis: Theory, Methods & Applications》2010,72(1):143-150
We analyze boundary value problems prescribing Dirichlet or Neumann boundary conditions for a nonlocal nonlinear diffusion operator which is analogous to the porous medium equation in a bounded smooth domain Ω∈RN with N≥1. First, we prove existence and uniqueness of solutions and the validity of a comparison principle for these problems. Next, we impose boundary data that blow up in finite time and study the behavior of the solutions. 相似文献
16.
Jinghua Wang 《Journal of Differential Equations》2003,189(1):1-16
In this paper, we study a generalized Burgers equation ut+(u2)x=tuxx, which is a non-uniformly parabolic equation for t>0. We show the existence and uniqueness of classical solutions to the initial-value problem of the generalized Burgers equation with rough initial data belonging to . 相似文献
17.
Ross G. Pinsky 《Journal of Differential Equations》2006,220(2):407-433
Consider classical solutions u∈C2(Rn×(0,∞))∩C(Rn×[0,∞)) to the parabolic reaction diffusion equation
18.
Goro Akagi 《Journal of Differential Equations》2011,251(7):1790-1812
We discuss the existence of periodic solution for the doubly nonlinear evolution equation A(u′(t))+∂?(u(t))∋f(t) governed by a maximal monotone operator A and a subdifferential operator ∂? in a Hilbert space H. As the corresponding Cauchy problem cannot be expected to be uniquely solvable, the standard approach based on the Poincaré map may genuinely fail. In order to overcome this difficulty, we firstly address some approximate problems relying on a specific approximate periodicity condition. Then, periodic solutions for the original problem are obtained by establishing energy estimates and by performing a limiting procedure. As a by-product, a structural stability analysis is presented for the periodic problem and an application to nonlinear PDEs is provided. 相似文献
19.
Zhichun Zhai 《Nonlinear Analysis: Theory, Methods & Applications》2010,72(6):3173-3189
In this paper, we study Keller-Segel systems with fractional diffusion and a nonlocal term. We establish the global existence, uniqueness and stability of solutions for systems with small initial data in critical Besov spaces. Our main tools are the Lp−Lq estimates for in Besov spaces and the perturbation of linearization. 相似文献
20.
The local well-posedness for the generalized two-dimensional (2D) Ginzburg-Landau equation is obtained for initial data in Hs(R2)(s>1/2). The global result is also obtained in Hs(R2)(s>1/2) under some conditions. The results on local and global well-posedness are sharp except the endpoint s=1/2. We mainly use the Tao's [k;Z]-multiplier method to obtain the trilinear and multilinear estimates. 相似文献