A diffusive Holling–Tanner prey–predator model with free boundary

2018 ◽  
Vol 11 (05) ◽  
pp. 1850066 ◽  
Author(s):  
Chenglin Li
Keyword(s):  
Boundary Condition ◽  
Free Boundary ◽  
Global Existence ◽  
Comparison Principle ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Threshold Value ◽  
Spreading Coefficient ◽  
Initial Size ◽  
Predator Model

This paper is concerned with a diffusive Holling–Tanner prey–predator model in a bounded domain with Dirichlet boundary condition and a free boundary. The global existence of the unique solution is proved. Moreover, the criteria governing spreading–vanishing are derived by mainly using the comparison principle. The results show that if the length of the occupying line is bigger than a threshold value (spreading barrier), then the spreading of predators will make an achievement, and, if the length of the occupying line is smaller than this spreading barrier and the spreading coefficient is relatively small depending on initial size of predators, then the predators will fail in establishing themselves and eventually die out.

scholarly journals Global Existence and General Decay of Solutions for a Quasilinear System with Degenerate Damping Terms

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Fatma Ekinci ◽  
Erhan Pișkin ◽  
Salah Mahmoud Boulaaras ◽  
Ibrahim Mekawy
Keyword(s):  
Boundary Condition ◽  
Global Existence ◽  
Recent Work ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
General Decay ◽  
Quasilinear System ◽  
Relaxation Functions ◽  
Decay Of Solutions ◽  
Viscoelastic Equations

In this work, we consider a quasilinear system of viscoelastic equations with degenerate damping, dispersion, and source terms under Dirichlet boundary condition. Under some restrictions on the initial datum and standard conditions on relaxation functions, we study global existence and general decay of solutions. The results obtained here are generalization of the previous recent work.

scholarly journals Parabolic equations with natural growth approximated by nonlocal equations

2020 ◽  
Vol 23 (01) ◽  
pp. 1950088
Author(s):  
Tommaso Leonori ◽  
Alexis Molino ◽  
Sergio Segura de León
Keyword(s):  
Boundary Condition ◽  
Asymptotic Behavior ◽  
Comparison Principle ◽  
Dirichlet Boundary Condition ◽  
Parabolic Equations ◽  
Dirichlet Boundary ◽  
Natural Growth ◽  
Nonlocal Problems ◽  
Nonlocal Equations ◽  
Bounded Smooth

In this paper, we study several aspects related with solutions of nonlocal problems whose prototype is [Formula: see text] where we take, as the most important instance, [Formula: see text] with [Formula: see text] as well as [Formula: see text], [Formula: see text] is a smooth symmetric function with compact support and [Formula: see text] is either a bounded smooth subset of [Formula: see text], with nonlocal Dirichlet boundary condition, or [Formula: see text] itself. The results deal with existence, uniqueness, comparison principle and asymptotic behavior. Moreover, we prove that if the kernel is rescaled in a suitable way, the unique solution of the above problem converges to a solution of the deterministic Kardar–Parisi–Zhang equation.

scholarly journals On some classes of generalized Schrödinger equations

2020 ◽  
Vol 10 (1) ◽  
pp. 522-533
Author(s):  
Amanda S. S. Correa Leão ◽  
Joelma Morbach ◽  
Andrelino V. Santos ◽  
João R. Santos Júnior
Keyword(s):  
Boundary Condition ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Laplacian Operator ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nontrivial Solutions ◽  
Homogeneous Dirichlet Boundary Condition ◽  
Stationary Problems

Abstract Some classes of generalized Schrödinger stationary problems are studied. Under appropriated conditions is proved the existence of at least 1 + $\begin{array}{} \sum_{i=2}^{m} \end{array}$ dim Vλi pairs of nontrivial solutions if a parameter involved in the equation is large enough, where Vλi denotes the eigenspace associated to the i-th eigenvalue λi of laplacian operator with homogeneous Dirichlet boundary condition.

On the Dispersive Estimate for the Dirichlet Schrödinger Propagator and Applications to Energy Critical NLS

2014 ◽  
Vol 66 (5) ◽  
pp. 1110-1142
Author(s):  
Dong Li ◽  
Guixiang Xu ◽  
Xiaoyi Zhang
Keyword(s):  
Boundary Condition ◽  
Fundamental Solution ◽  
Unit Ball ◽  
Dirichlet Boundary Condition ◽  
Obstacle Problem ◽  
Dirichlet Boundary ◽  
Radial Symmetry ◽  
Well Posedness ◽  
Robust Algorithm ◽  
Dispersive Estimate

AbstractWe consider the obstacle problem for the Schrödinger evolution in the exterior of the unit ball with Dirichlet boundary condition. Under radial symmetry we compute explicitly the fundamental solution for the linear Dirichlet Schrödinger propagator and give a robust algorithm to prove sharp L1 → L∞ dispersive estimates. We showcase the analysis in dimensions n = 5, 7. As an application, we obtain global well–posedness and scattering for defocusing energy-critical NLS on with Dirichlet boundary condition and radial data in these dimensions.

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