244 research outputs found

    Asymptotic behaviour of large solutions of quasilinear elliptic problems

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    The paper deals with the large solutions of the problems ā–³u=up\triangle u=u^p and ā–³u=eu.\triangle u= e^u. They blow up at the boundary. It is well-known that the first term in their asymptotic behaviour near the boundary is independent of the geometry of the boundary. We determine the second term which depends on the mean curvature of the nearest point on the boundary. The computation is based on suitable upper and lower solutions and on estimates given in [4]. In the last section these estimates are used together with the P-function to establish the asymptotic behaviour of the gradient

    Blow-up behavior of collocation solutions to Hammerstein-type volterra integral equations

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    We analyze the blow-up behavior of one-parameter collocation solutions for Hammerstein-type Volterra integral equations (VIEs) whose solutions may blow up in finite time. To approximate such solutions (and the corresponding blow-up time), we will introduce an adaptive stepsize strategy that guarantees the existence of collocation solutions whose blow-up behavior is the same as the one for the exact solution. Based on the local convergence of the collocation methods for VIEs, we present the convergence analysis for the numerical blow-up time. Numerical experiments illustrate the analysis

    Viewing the Steklov eigenvalues of the Laplace operator as critical Neumann eigenvalues

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    We consider the Steklov eigenvalues of the Laplace operator as limiting Neumann eigenvalues in a problem of boundary mass concentration. We discuss the asymptotic behavior of the Neumann eigenvalues in a ball and we deduce that the Steklov eigenvalues minimize the Neumann eigenvalues. Moreover, we study the dependence of the eigenvalues of the Steklov problem upon perturbation of the mass density and show that the Steklov eigenvalues violates a maximum principle in spectral optimization problems.Comment: This is a preprint version of a paper that will appear in the Proceedings of the 9th ISAAC Congress, Krak\'ow 201

    On the stability of solutions of semilinear elliptic equations with Robin boundary conditions on Riemannian manifolds

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    We investigate existence and nonexistence of stationary stable nonconstant solutions, i.e. patterns, of semilinear parabolic problems in bounded domains of Riemannian manifolds satisfying Robin boundary conditions. These problems arise in several models in applications, in particular in Mathematical Biology. We point out the role both of the nonlinearity and of geometric objects such as the Ricci curvature of the manifold, the second fundamental form of the boundary of the domain and its mean curvature. Special attention is devoted to surfaces of revolution and to spherically symmetric manifolds, where we prove refined results

    On Uniqueness of Boundary Blow-up Solutions of a Class of Nonlinear Elliptic Equations

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    We study boundary blow-up solutions of semilinear elliptic equations Lu=u+pLu=u_+^p with p>1p>1, or Lu=eauLu=e^{au} with a>0a>0, where LL is a second order elliptic operator with measurable coefficients. Several uniqueness theorems and an existence theorem are obtained.Comment: To appear in Comm. Partial Differential Equations; 10 page

    Quenching and Propagation of Combustion Without Ignition Temperature Cutoff

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    We study a reaction-diffusion equation in the cylinder Ī©=RƗTm\Omega = \mathbb{R}\times\mathbb{T}^m, with combustion-type reaction term without ignition temperature cutoff, and in the presence of a periodic flow. We show that if the reaction function decays as a power of TT larger than three as Tā†’0T\to 0 and the initial datum is small, then the flame is extinguished -- the solution quenches. If, on the other hand, the power of decay is smaller than three or initial datum is large, then quenching does not happen, and the burning region spreads linearly in time. This extends results of Aronson-Weinberger for the no-flow case. We also consider shear flows with large amplitude and show that if the reaction power-law decay is larger than three and the flow has only small plateaux (connected domains where it is constant), then any compactly supported initial datum is quenched when the flow amplitude is large enough (which is not true if the power is smaller than three or in the presence of a large plateau). This extends results of Constantin-Kiselev-Ryzhik for combustion with ignition temperature cutoff. Our work carries over to the case Ī©=RnƗTm\Omega = \mathbb{R}^n\times\mathbb{T}^m, when the critical power is 1+2/n1 + 2/n, as well as to certain non-periodic flows

    Maximizing Neumann fundamental tones of triangles

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    We prove sharp isoperimetric inequalities for Neumann eigenvalues of the Laplacian on triangular domains. The first nonzero Neumann eigenvalue is shown to be maximal for the equilateral triangle among all triangles of given perimeter, and hence among all triangles of given area. Similar results are proved for the harmonic and arithmetic means of the first two nonzero eigenvalues

    Positive solutions to indefinite Neumann problems when the weight has positive average

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    We deal with positive solutions for the Neumann boundary value problem associated with the scalar second order ODE u"+q(t)g(u)=0,tāˆˆ[0,T], u" + q(t)g(u) = 0, \quad t \in [0, T], where g:[0,+āˆž[ā€‰ā†’Rg: [0, +\infty[\, \to \mathbb{R} is positive on ā€‰]0,+āˆž[ā€‰\,]0, +\infty[\, and q(t)q(t) is an indefinite weight. Complementary to previous investigations in the case āˆ«0Tq(t)<0\int_0^T q(t) < 0, we provide existence results for a suitable class of weights having (small) positive mean, when gā€²(x)<0g'(x) < 0 at infinity. Our proof relies on a shooting argument for a suitable equivalent planar system of the type xā€²=y,yā€²=h(x)y2+q(t), x' = y, \qquad y' = h(x)y^2 + q(t), with h(x)h(x) a continuous function defined on the whole real line.Comment: 17 pages, 3 figure
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