scholarly journals Anisotropic nonlinear diffusion with absorption: existence and extinction

1996 ◽  
Vol 19 (3) ◽  
pp. 427-434 ◽  
Author(s):  
Alan V. Lair ◽  
Mark E. Oxley
Keyword(s):  
Nonlinear Diffusion ◽  
Dirichlet Boundary ◽  
Sufficient Conditions ◽  
Generalized Solution ◽  
Dirichlet Boundary Conditions ◽  
Parabolic Partial Differential Equation ◽  
Initial Condition ◽  
Nonlinear Parabolic ◽  
Necessary And Sufficient ◽  
Homogeneous Dirichlet Boundary Conditions

The authors prove that the nonlinear parabolic partial differential equation∂u∂t=∑i,j=1n∂2∂xi∂xjφij(u)−f(u)with homogeneous Dirichlet boundary conditions and a nonnegative initial condition has a nonnegative generalized solutionu. They also give necessary and sufficient conditions on the constitutive functionsφijandfwhich ensure the existence of a timet0>0for whichuvanishes for allt≥t0.

scholarly journals On the existence of dead cores for degenerate Lotka—Volterra models

2000 ◽  
Vol 130 (4) ◽  
pp. 743-766 ◽  
Author(s):  
M. Delgado ◽  
A. Suárez
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Diffusion ◽  
Positive Measure ◽  
Dirichlet Boundary ◽  
Sufficient Conditions ◽  
Dirichlet Boundary Conditions ◽  
Qualitative Properties ◽  
Linear Diffusion ◽  
Volterra Models ◽  
Homogeneous Dirichlet Boundary Conditions

In this work we study the existence and qualitative properties of non-negative solutions of the Lotka—Volterra models with nonlinear diffusion under homogeneous Dirichlet boundary conditions. We consider the three typical interactions: prey—predator, competition and symbiosis. Unlike the linear diffusion models, non-trivial non-negative solutions can exist which are not strictly positive. Sufficient conditions in terms of the coefficients involved in the setting of the models are given, assuring that one species (or both) does not survive on a set of its habitat (called ‘dead core’) of positive measure.

Extinction in a finite time for solutions of a class of quasilinear parabolic equations

2021 ◽  
pp. 1-23
Author(s):  
Y. Belaud ◽  
A. Shishkov
Keyword(s):  
Parabolic Equations ◽  
Finite Time ◽  
Dirichlet Boundary ◽  
Sufficient Conditions ◽  
Dirichlet Boundary Conditions ◽  
Power Functions ◽  
Quasilinear Parabolic Equations ◽  
Nonnegative Solutions ◽  
Finite Time Extinction ◽  
Necessary And Sufficient

We study the property of extinction in a finite time for nonnegative solutions of 1 q ∂ ∂ t ( u q ) − ∇ ( | ∇ u | p − 2 ∇ u ) + a ( x ) u λ = 0 for the Dirichlet Boundary Conditions when q > λ > 0, p ⩾ 1 + q, p ⩾ 2, a ( x ) ⩾ 0 and Ω a bounded domain of R N ( N ⩾ 1). We prove some necessary and sufficient conditions. The threshold is for power functions when p > 1 + q while finite time extinction occurs for very flat potentials a ( x ) when p = 1 + q.

scholarly journals Multiple solutions of fourth-order difference equations with different boundary conditions

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yuhua Long ◽  
Shaohong Wang ◽  
Jiali Chen
Keyword(s):  
Boundary Conditions ◽  
Fourth Order ◽  
Difference Equations ◽  
Multiple Solutions ◽  
Dirichlet Boundary ◽  
Sufficient Conditions ◽  
Dirichlet Boundary Conditions ◽  
Invariant Sets ◽  
Mountain Pass Lemma ◽  
Order Difference

Abstract In the present paper, a class of fourth-order nonlinear difference equations with Dirichlet boundary conditions or periodic boundary conditions are considered. Based on the invariant sets of descending flow in combination with the mountain pass lemma, we establish a series of sufficient conditions on the existence of multiple solutions for these boundary value problems. In addition, some examples are provided to demonstrate the applicability of our results.

scholarly journals Uniqueness of renormalized solutions to nonlinear parabolic problems with lower-order terms

2013 ◽  
Vol 143 (6) ◽  
pp. 1185-1208 ◽  
Author(s):  
Rosaria Di Nardo ◽  
Filomena Feo ◽  
Olivier Guibé
Keyword(s):  
Parabolic Equations ◽  
Dirichlet Boundary ◽  
Growth Conditions ◽  
Dirichlet Boundary Conditions ◽  
Parabolic Problems ◽  
Renormalized Solutions ◽  
Uniqueness Results ◽  
Nonlinear Parabolic ◽  
Image Position ◽  
Lower Order Terms

We consider a general class of parabolic equations of the typewith Dirichlet boundary conditions and with a right-hand side belonging to L1 + Lp′ (W−1, p′). Using the framework of renormalized solutions we prove uniqueness results under appropriate growth conditions and Lipschitz-type conditions on a(u, ∇u), K(u) and H(∇u).

A Lane–Emden system with singular data

2011 ◽  
Vol 141 (6) ◽  
pp. 1279-1294 ◽  
Author(s):  
Marius Ghergu
Keyword(s):  
Boundary Conditions ◽  
Elliptic System ◽  
Bounded Domain ◽  
Existence And Uniqueness ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Uniqueness Of Solutions ◽  
Homogeneous Dirichlet Boundary Conditions ◽  
Singular Data

We study the elliptic system −Δu = δ(x)−avp in Ω, −Δv = δ(x)−buq in Ω, subject to homogeneous Dirichlet boundary conditions. Here, Ω ⊂ ℝN, N ≥ 1, is a smooth and bounded domain, δ(x) = dist(x, ∂Ω), a, b ≥ 0 and p, q ∈ ℝ satisfy pq > −1. The existence, non-existence and uniqueness of solutions are investigated in terms of a, b, p and q.

Nonlinear Dirichlet problem with non local regional diffusion

2016 ◽  
Vol 19 (2) ◽  
Author(s):  
César E. Torres Ledesma
Keyword(s):  
Dirichlet Problem ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Nonlinear Term ◽  
Existence Of Solutions ◽  
Dirichlet Boundary Conditions ◽  
Nonlinear Dirichlet Problem ◽  
Non Local ◽  
Homogeneous Dirichlet Boundary Conditions

AbstractThe purpose of this paper is to study the existence of solutions for equations driven by a non-local regional operator with homogeneous Dirichlet boundary conditions. More precisely, we consider the problemwhere the nonlinear term

LEAST ENERGY NODAL SOLUTIONS OF THE BREZIS–NIRENBERG PROBLEM IN DIMENSION N = 5

2009 ◽  
Vol 11 (01) ◽  
pp. 59-69 ◽  
Author(s):  
PAOLO ROSELLI ◽  
MICHEL WILLEM
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
First Eigenvalue ◽  
Nodal Solutions ◽  
The First Eigenvalue ◽  
Homogeneous Dirichlet Boundary Conditions ◽  
Sign Changing Solutions ◽  
Nirenberg Problem ◽  
Least Energy

We prove the existence of (a pair of) least energy sign changing solutions of [Formula: see text] when Ω is a bounded domain in ℝN, N = 5 and λ is slightly smaller than λ1, the first eigenvalue of -Δ with homogeneous Dirichlet boundary conditions on Ω.

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