1,721,069 research outputs found
Global Solutions of Semilinear Parabolic Equations on Negatively Curved Riemannian Manifolds
Full text linkWe are concerned with global existence for semilinear parabolic equations on Riemannian manifolds with negative sectional curvatures. A particular attention is paid to the class of initial conditions which ensure existence of global solutions. Indeed, we show that such a class is crucially related to the curvature bounds. A crucial point in our arguments is the construction of positive bounded supersolutions to the eigenvalue equation
Integral conditions for uniqueness of solutions to degenerate parabolic equations
No full textWe investigate uniqueness of solutions to the initial value problem for degenerate parabolic equations, posed in bounded domains, where no boundary conditions are prescribed. In order to obtain uniqueness, we need that the solutions satisfy certain integral growth conditions, which are crucially related to the degeneracy of the operator near the boundary. In particular, such solutions can be unbounded near the boundary. Our hypothesis on the behavior of the operator at the boundary is optimal; in fact, we show that if it fails, then nonuniqueness of solutions prevails
GLOBAL SOLUTIONS OF SEMILINEAR PARABOLIC EQUATIONS WITH DRIFT TERM ON RIEMANNIAN MANIFOLDS
Full text linkWe study existence and non-existence of global solutions to the semilinear heat equation with a drift term and a power-like source term u(P), on Cartan-Hadamard manifolds. Under suitable assumptions on Ricci and sectional curvatures, we show that, for any p > 1, global solutions cannot exists if the initial datum is large enough. Furthermore, under appropriate conditions on the drift term, global existence is obtained for any p > 1, if the initial datum is sufficiently small. We also deal with Riemannian manifolds whose Ricci curvature tends to zero at infinity sufficiently fast. We show that for any non trivial initial datum, for certain p depending on the Ricci curvature bound, global solutions cannot exist. On the other hand, for certain values of p, depending on the vector field b, global solutions exist, for sufficiently small initial data
Weighted Poincare Inequalities and Degenerate Elliptic and Parabolic Problems: An Approach via the Distance Function
No full textWe obtain weighted Poincare inequalities in bounded domains, where the weight is given by a symmetric nonnegative definite matrix, which can degenerate on submanifolds. Furthermore, we investigate uniqueness and nonuniqueness of solutions to degenerate elliptic and parabolic problems, where the diffusion matrix can degenerate on subsets of the boundary of the domain. Both the results are obtained by means of the distance function from the degeneracy set, which is used to construct suitable local sub- and supersolutions
Blow-up for semilinear parabolic equations in cones of the hyperbolic space
Full text linkWe investigate existence and nonexistence of global in time nonnegative solutions to the semilinear heat equation, with a reaction term of the type e(mu t)u(p) (,u is an element of R,p > 1), posed on cones of the hyperbolic space. Under a certain assumption on ,u and p, related to the bottom of the spectrum of -triangle in H-n, we prove that any solution blows up in finite time, for any nontrivial nonnegative initial datum. Instead, if the parameters ,u and p satisfy the opposite condition we have: (a) blow-up when the initial datum is large enough, (b) existence of global solutions when the initial datum is small enough. Hence our conditions on the parameters ,u and p are optimal. We see that blow-up and global existence do not depend on the amplitude of the cone. This is very different from what happens in the Euclidean setting (Bandle and Levine 1989 Trans. Am. Math. Soc. 316 595-622), and it is essentially due to a specific geometric feature of H-n
Blow-up and global existence for solutions to the porous medium equation with reaction and slowly decaying density
Full text linkWe study existence of global solutions and finite time blow-up of solutions to the Cauchy problem for the porous medium equation with a variable density ρ(x) and a power-like reaction term ρ(x)up with p>1; this is a mathematical model of a thermal evolution of a heated plasma (see [29]). The density decays slowly at infinity, in the sense that ρ(x)≲|x|−q as |x|→+∞ with q∈[0,2). We show that for large enough initial data, solutions blow-up in finite time for any p>1. On the other hand, if the initial datum is small enough and p>p ̄, for a suitable p ̄ depending on ρ,m,N, then global solutions exist. In addition, if p0 small enough, when m≤
A Liouville-type theorem for elliptic equations with singular coefficients in bounded domains
No full textWe investigate Liouville-type theorems for elliptic equations with a drift and with a potential posed in bounded domains. We provide sufficient conditions on the potential and on the drift term in order to the equation does not admit nontrivial bounded solutions. We also show that such conditions are optimal. Indeed, when they fail, the elliptic equation possesses infinitely many bounded solutions
BLOW-UP ON METRIC GRAPHS AND RIEMANNIAN MANIFOLDS
No full textWe study blow-up versus global existence of solutions to a model semilinear parabolic equation in metric measure spaces. Applications to metric graphs and Riemannian manifolds are considered, pointing out the occurrence of the Fujita phenomenon
Blow-up and global existence for the inhomogeneous porous medium equation with reaction
Full text linkWe study finite time blow-up and global existence of solutions to the Cauchy problem for the porous medium equation with a variable density ρ(x) and a power-like reaction term. We firstly consider the case that ρ(x) decays at infinity like the critical case [x]-2divided by a positive power of the logarithm of [x] and we show that for small enough initial data, solutions globally exist for any p > 1. On the other hand, when ρ(x) decays at infinity like the critical case [x]-2multiplied by a positive power of the logarithm of [x], if the initial datum is small enough, then one has global existence of the solution for any p > m, while if the initial datum is large enough, then the blow-up of the solutions occurs for any p > m. Such results generalize those established in [27] and [28], where it is supposed that ρ(x) decays at infinity like a power of [x], without logarithmic terms
UNIQUENESS FOR FRACTIONAL PARABOLIC AND ELLIPTIC EQUATIONS WITH DRIFT
No full textWe investigate uniqueness, in suitable weighted Lebesgue spaces, of solutions to a class of linear, nonlocal parabolic problems with a drift. More precisely, the problem is nonlocal due to the presence of the fractional Laplacian as diffusion operator. The drift term is driven by a smooth enough, possibly unbounded vector field b which satisfies a suitable growth condition in the set {x ∈ RN : > 0}. In general, our uniqueness class includes unbounded solutions; in particular, we get uniqueness of bounded solutions. Furthermore, we show sharpness of the hypothesis on the drift term b; in fact we show that, if the drift term b violates, in an appropriate sense, the mentioned growth condition (see (2.5)), then infinitely many bounded solutions to the problem exist. Finally, we also investigate uniqueness of a linear, nonlocal elliptic equation with a drift term obtaining similar results
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