Free boundary problems with local–nonlocal diffusions and different free boundaries II: Spreading–vanishing and long-time behavior

2022 ◽  
Vol 64 ◽  
pp. 103445
Author(s):  
Xiu Dong ◽  
Jianping Wang ◽  
Mingxin Wang
Keyword(s):  
Free Boundary ◽  
Free Boundary Problems ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Free Boundaries ◽  
Boundary Problems ◽  
Long Time

scholarly journals Free boundaries in problems with hysteresis

2015 ◽  
Vol 373 (2050) ◽  
pp. 20140271 ◽  
Author(s):  
D. E. Apushkinskaya ◽  
N. N. Uraltseva
Keyword(s):  
Free Boundary ◽  
Sobolev Class ◽  
Free Boundary Problems ◽  
Strong Solutions ◽  
Chemical Processes ◽  
Free Boundaries ◽  
Open Problems ◽  
Boundary Problems ◽  
Hysteresis Operator ◽  
With Memory

Here, we present a survey concerning parabolic free boundary problems involving a discontinuous hysteresis operator. Such problems describe biological and chemical processes ‘with memory’ in which various substances interact according to hysteresis law. Our main objective is to discuss the structure of the free boundaries and the properties of the so-called ‘strong solutions’ belonging to the anisotropic Sobolev class with sufficiently large q . Several open problems in this direction are proposed as well.

Unsteady two-dimensional flows with free boundaries. - I. General theory

1956 ◽  
Vol 235 (1202) ◽  
pp. 375-381 ◽  
Keyword(s):  
Free Boundary ◽  
Free Boundary Problems ◽  
Steady State Solution ◽  
Two Dimensional ◽  
Free Boundaries ◽  
Boundary Problems ◽  
Initial Development ◽  
Bernoulli’S Equation ◽  
Hodograph Plane ◽  
Two Dimensional Flows

The paper contains a summary of relevant earlier work on free-boundary problems (§1) and then considers the initial development of steady two-dimensional flows. The motion of an incompressible in viscid fluid with free boundaries is considered (§2) by transforming into the hodograph plane (In q 0 / U , θ 0 ) of the steady flow. The equation of the free boundary and the velocity potential are expanded in powers of e- λt . Thus ϕ ( x, y, t ) = ϕ 0 ( x, y ) + e - λt ϕ 1 ( x, y ) + ..., where ϕ 0 is the known steady-state solution and ϕ 1 is to be determined. The exact boundary condition, which is the unsteady form of Bernoulli’s equation, is applied on the free boundary which is not taken as a streamline. A general discussion of the validity of the approach is given (§3). It is foreshadowed that for jet flow through a slit the predicted shape of the jet will probably have a kink at the nose; this is consistent with the assumptions made in the analysis.

scholarly journals Two-Phase Free Boundary Problems: From Existence to Smoothness

2017 ◽  
Vol 17 (2) ◽  
Author(s):  
Daniela De Silva ◽  
Fausto Ferrari ◽  
Sandro Salsa
Keyword(s):  
Free Boundary ◽  
Viscosity Solutions ◽  
Free Boundary Problems ◽  
Elliptic Operators ◽  
Free Boundaries ◽  
Two Phase ◽  
Boundary Problems ◽  
Open Questions

AbstractWe describe the theory we developed in recent times concerning two-phase free boundary problems governed by elliptic operators with forcing terms. Our results range from existence of viscosity solutions to smoothness of both solutions and free boundaries. We also discuss some open questions, possible object of future investigation.

scholarly journals $m$th-order Fisher-KPP equation with free boundaries and time-aperiodic advection

2021 ◽  
Author(s):  
Changqing Ji ◽  
Dandan Zhu ◽  
Jingli Ren
Keyword(s):  
Large Amplitude ◽  
Small Amplitude ◽  
Initial Conditions ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Free Boundaries ◽  
Kpp Equation ◽  
Advection Term ◽  
Long Time ◽  
Theoretical Results

In this paper, we investigate a $m$th-order Fisher-KPP equation with free boundaries and time-aperiodic advection. Considering the influence of advection term and initial conditions on the long time behavior of solutions, we obtain spreading-vanishing dichotomy, spreading-transition-vanishing trichotomy, and vanishing happens with the coefficient of advection term in small amplitude, medium-sized amplitude and large amplitude, respectively. Then, the appropriate parameters are selected in the simulation to intuitively show the corresponding theoretical results. Moreover, the wave-spreading and wave-vanishing cases of the solutions are observed in our study.

Sign in / Sign up

Export Citation Format

Share Document