Global existence for a Leibenson type equation with reaction on Riemannian manifolds

2025/05/13 by Meglioli, Giulia, Oliva, Francescantonio, Petitta, Francesco · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2505.08304

Abstract

We show a global existence result for a doubly nonlinear porous medium type equation of the form u_t = Δ_p u^m + u^q on a complete and non-compact Riemannian manifold M of infinite volume. Here, for 11 and q>m(p-1). In particular, under the assumptions that M supports the Sobolev inequality, we prove that a solution for such a problem exists globally in time provided q>m(p-1)+\frac pN and the initial datum is small enough; namely, we establish an explicit bound on the L^∞ norm of the solution at all positive times, in terms of the L¹ norm of the data. Under the additional assumption that a Poincaré-type inequality also holds in M, we can establish the same result in the larger interval, i.e. q>m(p-1). This result has no Euclidean counterpart, as it differs entirely from the case of a bounded Euclidean domain due to the fact that M is non-compact and has infinite measure.

Citations

• Nonexistence of Solutions to classes of parabolic inequalities in the Riemannian setting
• Global existence for reaction-diffusion evolution equations driven by the p-Laplacian on manifolds
• Global existence and blow-up of solutions to the porous medium equation with reaction and singular coefficients
• Blow-up versus global existence of solutions for reaction-diffusion equations on classes of Riemannian manifolds
• Nonexistence of solutions to quasilinear parabolic equations with a potential in bounded domains
• Global existence of solutions and smoothing effects for classes of\n reaction-diffusion equations on manifolds
• Blow-up and global existence for the inhomogeneous porous medium equation with reaction
• Smoothing effects and infinite time blowup for reaction-diffusion\n equations: an approach via Sobolev and Poincar 'e inequalities
• Blow-up and global existence for solutions to the porous medium equation with reaction and slowly decaying density
• Blow-up and global existence for solutions to the porous medium equation with reaction and fast decaying density
• Blow-up and global existence for the porous medium equation with\n reaction on a class of Cartan-Hadamard manifolds
• The porous medium equation with measure data on negatively curved\n Riemannian manifolds
• Smoothing effects for the porous medium equation on Cartan-Hadamard\n manifolds
• Nonexistence of solutions to parabolic differential inequalities with a potential on Riemannian manifolds

Cited by

• Gradient estimates for Leibenson's equation on Riemannian manifolds

Related

• Blow-up and global existence for the porous medium equation with\n reaction on a class of Cartan-Hadamard manifolds
• Global existence of solutions and smoothing effects for classes of\n reaction-diffusion equations on manifolds
• Blow-up and global existence for the inhomogeneous porous medium equation with reaction
• Global existence for reaction-diffusion evolution equations driven by the p-Laplacian on manifolds
• The porous medium equation with large data on Cartan-Hadamard manifolds under general curvature bounds