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1.
We study the existence of singular solutions to the equation −div(|Du|p−2Du)=|u|q−1u under the form u(r,θ)=rβω(θ), r>0, θSN−1. We prove the existence of an exponent q below which no positive solutions can exist. If the dimension is 2 we use a dynamical system approach to construct solutions.  相似文献   

2.
3.
We start by studying the existence of positive solutions for the differential equation
u=a(x)ug(u),  相似文献   

4.
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−2u), with initial data u0Lr(Ω) is proved under r>N(qp)/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   

5.
The paper discusses the existence of positive and dead core solutions of the singular differential equation (?(u))=λf(t,u,u,u) satisfying the boundary conditions u(0)=A, u(T)=A, min{u(t):t∈[0,T]}=0. Here λ is a nonnegative parameter, A is a positive constant and the Carathéodory function f(t,x,y,z) is singular at the value 0 of its space variable y.  相似文献   

6.
7.
We consider, for p∈(1,2) and q>1, self-similar singular solutions of the equation vt=div(|∇v|p−2v)−vq in Rn×(0,∞); here by self-similar we mean that v takes the form v(x,t)=tαw(|x|tαβ) 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   

8.
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   

9.
10.
We discuss the existence of global or periodic solutions to the nonlinear wave equation with the boundary condition , where Ω is a bounded domain in RN,ρ(x,v) is a function like ρ(x,v)=a(x)g(v) with g′(v)?0 and β(x,u) is a source term of power nonlinearity. a(x) is assumed to be positive only in a neighborhood of a part of the boundary ∂Ω and the stability property is very delicate, which makes the problem interesting.  相似文献   

11.
带非局部源的退化半线性抛物方程的解的爆破性质   总被引:1,自引:0,他引:1  
This paper deals with the blow-up properties of the positive solutions to the nonlocal degenerate semilinear parabolic equation u t − (x a u x ) x =∫ 0 a f(u)dx in (0,a) × (0,T) under homogeneous Dirichlet conditions. The local existence and uniqueness of classical solution are established. Under appropriate hypotheses, the global existence and blow-up in finite time of positve solutions are obtained. It is also proved that the blow-up set is almost the whole domain. This differs from the local case. Furthermore, the blow-up rate is precisely determined for the special case: f(u)=u p , p>1.  相似文献   

12.
13.
We study the equation Δu+u|u|p−1+V(x)u+f(x)=0 in Rn, where n?3 and p>n/(n−2). The forcing term f and the potential V can be singular at zero, change sign and decay polynomially at infinity. We can consider anisotropic potentials of form h(x)|x|−2 where h is not purely angular. We obtain solutions u which blow up at the origin and do not belong to any Lebesgue space Lr. Also, u is positive and radial, in case f and V are. Asymptotic stability properties of solutions, their behavior near the singularity, and decay are addressed.  相似文献   

14.
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   

15.
In this paper we study the large time behavior of non-negative solutions to the Cauchy problem of utumuq in RN×(0,∞), where m>1 and q=qcm+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 utum, independent of the initial value, with additional logarithmic anomalous decay exponent in time as t→∞.  相似文献   

16.
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   

17.
The Dirichlet problem is considered for the heat equation ut=auxx, a>0 a constant, for (x,t)∈[0,1]×[0,T], without assuming any compatibility condition between initial and boundary data at the corner points (0,0) and (1,0). Under some smoothness restrictions on the data (stricter than those required by the classical maximum principle), weak and strong supremum and infimum principles are established for the higher-order derivatives, ut and uxx, of the bounded classical solutions. When compatibility conditions of zero order are satisfied (i.e., initial and boundary data coincide at the corner points), these principles allow to estimate the higher-order derivatives of classical solutions uniformly from below and above on the entire domain, except that at the two corner points. When compatibility conditions of the second order are satisfied (i.e., classical solutions belong to on the closed domain), the results of the paper are a direct consequence of the classical maximum and minimum principles applied to the higher-order derivatives. The classical principles for the solutions to the Dirichlet problem with compatibility conditions are generalized to the case of the same problem without any compatibility condition. The Dirichlet problem without compatibility conditions is then considered for general linear one-dimensional parabolic equations. The previous results as well as some new properties of the corresponding Green functions derived here allow to establish uniformL1-estimates for the higher-order derivatives of the bounded classical solutions to the general problem.  相似文献   

18.
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   

19.
We consider an Allen-Cahn type equation of the form utu+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
  相似文献   

20.
We study the smoothness properties of solutions to the coupled system of equations of Korteweg—de Vries type. We show that the equations dispersive nature leads to a gain in regularity for the solution. In particular, if the initial data (u0, v0 possesses certain regularity and sufficient decay as x → ∞, then the solution (u(t). v(t)) will be smoother than (u0, v0) for 0 < tT where T is the existence time of the solution.  相似文献   

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