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The proof of the Lane–Emden conjecture in four space dimensions
Abstract We partially solve a well-known conjecture about the nonexistence of positive entire solutions to elliptic systems of Lane–Emden type when the pair of exponents lies below the criticalExpand
Uniform Blow-Up Profiles and Boundary Behavior for Diffusion Equations with Nonlocal Nonlinear Source
In this paper, we introduce a new method for investigating the rate and profile of blow-up of solutions of diffusion equations with nonlocal nonlinear reaction terms. For large classes of equations,Expand
Sharp Gradient Estimate and Yau's Liouville Theorem for the Heat Equation on Noncompact Manifolds
We derive a sharp, localized version of elliptic type gradient estimates for positive solutions (bounded or not) to the heat equation. These estimates are related to the Cheng–Yau estimate for theExpand
Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States
Preliminaries.- Model Elliptic Problems.- Model Parabolic Problems.- Systems.- Equations with Gradient Terms.- Nonlocal Problems.
Liouville-type theorems and bounds of solutions of Hardy-Hénon equations
Abstract We consider the Hardy–Henon equation − Δ u = | x | a u p with p > 1 and a ∈ R and we are concerned in particular with the Liouville property, i.e. the nonexistence of positive solutions inExpand
Singularity and decay estimates in superlinear problems via Liouville-type theorems, I: Elliptic equations and systems
In this paper, we study some new connections between Liouville-type theorems and local properties of nonnegative solutions to superlinear elliptic problems. Namely, we develop a general method forExpand
Blow-up in nonlocal reaction-diffusion equations
We present new blow-up results for reaction-diffusion equations with nonlocal nonlinearities. The nonlocal source terms we consider are of several types, and are relevant to various models in physicsExpand
Singularity and decay estimates in superlinear problems via liouville-type theorems. Part II: Parabolic equations
In this paper, we study some new connections between parabolic Liouville-type theorems and local and global properties of nonnegative classical solutions to superlinear parabolic problems, with orExpand
Global solutions of inhomogeneous Hamilton-Jacobi equations
We consider the viscous Hamilton-Jacobi (VHJ) equationut-Δu=|∇u|p+h(x). For the Dirichlet problem withp>2, it is known thatgradient blow-up may occur in finite time (on the boundary). WhereasExpand
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