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1.
In this note, we investigate the regularity of the extremal solution u? for the semilinear elliptic equation −△u+c(x)⋅∇u=λf(u) on a bounded smooth domain of Rn with Dirichlet boundary condition. Here f is a positive nondecreasing convex function, exploding at a finite value a∈(0,∞). We show that the extremal solution is regular in the low-dimensional case. In particular, we prove that for the radial case, all extremal solutions are regular in dimension two.  相似文献   

2.
Let u? be a single layered radially symmetric unstable solution of the Allen-Cahn equation −?2Δu=u(ua(|x|))(1−u) over the unit ball with Neumann boundary conditions. Based on our estimate of the small eigenvalues of the linearized eigenvalue problem at u? when ? is small, we construct solutions of the form u?+v?, with v? non-radially symmetric and close to zero in the unit ball except near one point x0 such that |x0| is close to a nondegenerate critical point of a(r). Such a solution has a sharp layer as well as a spike.  相似文献   

3.
This paper deals with positive solutions of degenerate and quasilinear parabolic systems not in divergence form: ut=up(Δu+av), vt=vq(Δv+bu), with null Dirichlet boundary conditions and positive initial conditions, where p, q, a and b are all positive constants. The local existence and uniqueness of classical solution are proved. Moreover, it will be proved that all solutions exist globally if and only if ab?λ12, where λ1 is the first eigenvalue of −Δ in Ω with homogeneous Dirichlet boundary condition.  相似文献   

4.
This paper is concerned with the study of the FitzHugh-Nagumo equations. These equations arise in mathematical biology as a model of the transmission of electrical impulses through a nerve axon; they are a simplified version of the Hodgkin-Huxley equations. The FitzHugh-Nagumo equations consist of a non-linear diffusion equation coupled to an ordinary differential equation. vt = vxx + f(v) ? u, ut = σv ? γu. We study these equations with either Dirichlet or Neumann boundary conditions, proving local and global existence, and uniqueness of the solutions. Furthermore, we obtain L estimates for the solutions in terms of the L1 norm of the boundary data, when the boundary data vanish after a finite time and the initial data are zero. These estimates allow us to prove exponential decay of the solutions.  相似文献   

5.
In this paper, we prove the relation v(t)?u(t,x)?w(t), where u(t,x) is the solution of an impulsive parabolic equations under Neumann boundary condition ∂u(t,x)/∂ν=0, and v(t) and w(t) are solutions of two impulsive ordinary equations. We also apply these estimates to investigate the asymptotic behavior of a model in the population dynamics, and it is shown that there exists a unique solution of the model which converges to the periodic solution of an impulsive ordinary equation asymptotically.  相似文献   

6.
We consider three singularly perturbed convection-diffusion problems defined in three-dimensional domains: (i) a parabolic problem −?(uxx+uyy)+ut+v1ux+v2uy=0 in an octant, (ii) an elliptic problem −?(uxx+uyy+uzz)+v1ux+v2uy+v3uz=0 in an octant and (iii) the same elliptic problem in a half-space. We consider for all of these problems discontinuous boundary conditions at certain regions of the boundaries of the domains. For each problem, an asymptotic approximation of the solution is obtained from an integral representation when the singular parameter ?→0+. The solution is approximated by a product of two error functions, and this approximation characterizes the effect of the discontinuities on the small ?− behaviour of the solution and its derivatives in the boundary layers or the internal layers.  相似文献   

7.
We apply the Five Functionals Fixed Point Theorem to verify the existence of at least three positive pseudo-symmetric solutions for the three point boundary value problem, (g(u′))′+a(t)f(u)=0, u(0)=0, and u(ν)=u(1), where g(v)=|v|p−2v, with p>1 and ν∈(0,1).  相似文献   

8.
We establish a series of properties of symmetric, N-pulse, homoclinic solutions of the reduced Gray-Scott system: u=uv2, v=vuv2, which play a pivotal role in questions concerning the existence and self-replication of pulse solutions of the full Gray-Scott model. Specifically, we establish the existence, and study properties, of solution branches in the (α,β)-plane that represent multi-pulse homoclinic orbits, where α and β are the central values of u(x) and v(x), respectively. We prove bounds for these solution branches, study their behavior as α→∞, and establish a series of geometric properties of these branches which are valid throughout the (α,β)-plane. We also establish qualitative properties of multi-pulse solutions and study how they bifurcate, i.e., how they change along the solution branches.  相似文献   

9.
In this paper we consider the quasilinear elliptic system Δpu=uavb, Δpv=ucve in a smooth bounded domain ΩRN, with the boundary conditions u=v=+∞ on ∂Ω. The operator Δp stands for the p-Laplacian defined by Δpu=div(|∇u|p−2u), p>1, and the exponents verify a,e>p−1, b,c>0 and (ap+1)(ep+1)?bc. We analyze positive solutions in both components, providing necessary and sufficient conditions for existence. We also prove uniqueness of positive solutions in the case (ap+1)(ep+1)>bc and obtain the exact blow-up rate near the boundary of the solution. In the case (ap+1)(ep+1)=bc, infinitely many positive solutions are constructed.  相似文献   

10.
In this article, we consider uniqueness of positive radial solutions to the elliptic system Δu+a(|x|)f(u,v)=0, Δv+b(|x|)g(u,v)=0, subject to the Dirichlet boundary condition on the open unit ball in RN (N?2). Our uniqueness results applies to, for instance, f(u,v)=uqvp, g(u,v)=upvq, p,q>0, p+q<1 or more general cases.  相似文献   

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