1,721,074 research outputs found
Global existence for the nonlinear heat equation on riemannian manifolds with negative sectional curvature
No full textWe address local existence, blow-up and global existence of mild solutions to the semilinear heat equation on Riemannian manifolds with negative sectional curvature. We deal with a power nonlinearity multiplied by a time-dependent positive function h(t), and initial conditions u0∈Lp(M). We show that depending on the behavior at infinity of h, either every solution blows up in finite time, or a global solution exists, if the initial datum is small enough. In particular, for any power nonlinearity, if h≡1 we have global existence for small initial data, whereas if h(t)=eαt a Fujita type phenomenon prevails varying the parameter α>0
Uniqueness and non-uniqueness of solutions to quasilinear parabolic equations with a singular coefficient on weighted Riemannian manifolds
No full textWe study, on weighted Riemannian model manifolds, well posedness of the Cauchy problem for a class of quasilinear parabolic equations with a coefficient which can be singular at infinity. We establish uniqueness or non-uniqueness of bounded solutions, under suitable assumptions on the behavior at infinity of the singular coefficient and on the Green function for the weighted Laplace-Beltrami operator. ©2012-IOS Press and the authors. All rights reserved
On the Cauchy problem for nonlinear parabolic equations with variable density
No full textWe investigate the well-posedness of the Cauchy problem for a class of nonlinear parabolic equations with variable density. Sufficient conditions for uniqueness or nonuniqueness in L∞(IRN × (0, T)) (N ≥ 3) are established in dependence of the behavior of the density at infinity. We deal with conditions at infinity of Dirichlet type, and possibly inhomogeneous. © 2009 Birkhäuser Verlag Basel/Switzerland
Support properties of solutions to nonlinear parabolic equations with variable density in the hyperbolic space
No full textWe consider the Cauchy problem for a class of nonlinear parabolic equations with variable density in the hyperbolic space, assuming that the initial datum has compact support. We provide simple conditions, involving the behaviour of the density at infinity, so that the support of every nonnegative solution is not compact at some positive time, or it remains compact for any positive time. These results extend to the case of the hyperbolic space those given in [8] for the Cauchy problem in IRn
Uniqueness of solutions to degenerate parabolic and elliptic equations in weighted Lebesgue spaces
No full textWe investigate uniqueness for degenerate parabolic and elliptic equations in the class of solutions belonging to weighted Lebesgue spaces and not satisfying any boundary condition. The uniqueness result that we provide relies on the existence of suitable positive supersolutions of the adjoint equations. Under proper assumptions on the behavior at the boundary of the coefficients of the operator, such supersolutions are constructed, mainly using the distance function from the boundary. © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
The existence of patterns on surfaces of revolution without boundary
No full textWe give a sufficient condition for the existence of patterns on surfaces of revolution of R3 without boundary. Such a condition involves the Gauss curvature of the surface and the geodesic curvature of parallels. An analogous result for surfaces of revolution with boundary is established in Bandle et al. (2012) [4]. © 2012 Elsevier Ltd. All rights reserved
On well-posedness of the semilinear heat equation on the sphere
No full textWe are concerned with existence, uniqueness and nonuniqueness of nonnegative solutions to the semilinear heat equation in open subsets of the n-dimensional sphere. Existence and uniqueness results are obtained using L p → L q estimates for the semigroup generated by the Laplace-Beltrami operator. Moreover, under proper assumptions on the nonlinear function, we establish nonuniqueness of weak solutions, when n ≥ 3; to do this, we shall prove uniqueness of positive classical solutions and nonuniqueness of singular solutions of the corresponding semilinear elliptic problem. © 2012 Springer Basel AG
Uniqueness for the heat equation in Riemannian manifolds
No full textWe investigate uniqueness, in suitable weighted Lebesgue spaces, of solutions to the heat equation in geodesically complete Riemannian manifolds
Liouville theorems for fully nonlinear elliptic equations on spherically symmetric Riemannian manifolds
No full textWe prove Hadamard and Liouville type theorems for viscosity supersolutions to fully nonlinear elliptic equations on spherically symmetric complete noncompact Riemannian manifolds. © 2012 Springer Basel
Phragmèn-Lindelöf principles for fully nonlinear elliptic equations with unbounded coefficients
No full textWe investigate Phragmèn-Lindelöf principles for viscosity solutions of fully nonlinear elliptic equations with possibly unbounded coefficients
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