Results 31 to 40 of about 156 (42)

Rate of convergence for one-dimensional quasilinear parabolic problem and its applications [PDF]

arXiv, 2016
Based on a comparison principle, we derive an exponential rate of convergence for solutions to the initial-boundary value problem for a class of quasilinear parabolic equations in one space dimension. We then apply the result to some models in population dynamics and image processing.
arxiv  

Equivalence of viscosity and weak solutions for a $p$-parabolic equation [PDF]

arXiv, 2019
We study the relationship of viscosity and weak solutions to the equation \[ \smash{\partial_{t}u-\Delta_{p}u=f(Du)} \] where $p>1$ and $f\in C(\mathbb{R}^{N})$ satisfies suitable assumptions. Our main result is that bounded viscosity supersolutions coincide with bounded lower semicontinuous weak supersolutions.
arxiv  

On a weighted anisotropic eigenvalue problem [PDF]

Glasgow Math. J. 67 (2025) 72-85
In this paper we deal with a weighted eigenvalue problem for the anisotropic $(p,q)$-Laplacian with Dirichlet boundary conditions. We study the main properties of the first eigenvalue and prove a reverse H\"older type inequality for the corresponding eigenfunctions.
arxiv   +1 more source

A comparison theorem for nonsmooth nonlinear operators [PDF]

arXiv, 2019
We prove a comparison theorem for super- and sub-solutions with non-vanishing gradients to semilinear PDEs provided a nonlinearity $f$ is $L^p$ function with $p > 1$. The proof is based on a strong maximum principle for solutions of divergence type elliptic equations with VMO leading coefficients and with lower order coefficients from a Kato class.
arxiv  

Nonlinear elliptic-parabolic problems [PDF]

, 2012
We introduce a notion of viscosity solutions for a general class of elliptic-parabolic phase transition problems. These include the Richards equation, which is a classical model in filtration theory. Existence and uniqueness results are proved via the comparison principle. In particular, we show existence and stability properties of maximal and minimal
arxiv   +1 more source

A note on the complete rotational invariance of biradial solutions to semilinear elliptic equations [PDF]

arXiv, 2010
We investigate symmetry properties of solutions to equations of the form $$ -\Delta u = \frac{a}{|x|^2} u + f(|x|, u)$$ in R^N for $N \geq 4$, with at most critical nonlinearities. By using geometric arguments, we prove that solutions with low Morse index (namely 0 or 1) and which are biradial (i.e.
arxiv  

Viscosity Solutions for the two-phase Stefan Problem [PDF]

arXiv, 2010
We introduce a notion of viscosity solutions for the two-phase Stefan problem, which incorporates possible existence of a mushy region generated by the initial data. We show that a comparison principle holds between viscosity solutions, and investigate the coincidence of the viscosity solutions and the weak solutions defined via integration by parts ...
arxiv  

Optimal estimates from below for biharmonic Green functions [PDF]

arXiv, 2011
Optimal pointwise estimates are derived for the biharmonic Green function under Dirichlet boundary conditions in arbitrary $C^{4,\gamma}$-smooth domains. Maximum principles do not exist for fourth order elliptic equations and the Green function may change sign.
arxiv  

Population structured by a space variable and a phenotypical trait [PDF]

arXiv, 2011
We consider populations structured by a phenotypic trait and a space variable, in a non-homogeneous environment. In the case of sex- ual populations, we are able to derive models close to existing mod- els in theoretical biology, from a structured population model.
arxiv