A mathematical model for long waves generated by wavemakers in non-linear dispersive systems

1973 ◽  
Vol 73 (2) ◽  
pp. 391-405 ◽  
Author(s):  
J. L. Bona ◽  
P. J. Bryant
Keyword(s):  
Mathematical Model ◽  
Initial Data ◽  
Water Waves ◽  
Open Channel ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Finite Amplitude ◽  
Mathematical Properties ◽  
Image Position ◽  
Initial Boundary Value

An initial-boundary-value problem for the equationis considered for x, t ≥ 0. This system is a model for long water waves of small but finite amplitude, generated in a uniform open channel by a wavemaker at one end. It is shown that, in contrast to an alternative, more familiar model using the Korteweg–deVries equation, the solution of (a) has good mathematical properties: in particular, the problem is well set in Hadamard's classical sense that solutions corresponding to given initial data exist, are unique, and depend continuously on the specified data.

Asymptotic stability of spatial homogeneity in a haptotaxis model for oncolytic virotherapy

2021 ◽  
pp. 1-21
Author(s):  
Youshan Tao ◽  
Michael Winkler
Keyword(s):  
Initial Data ◽  
Blow Up ◽  
Reaction Diffusion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Oncolytic Virotherapy ◽  
Homogeneous Distribution ◽  
Spatial Homogeneity ◽  
Image Position ◽  
Initial Boundary Value

This study considers a model for oncolytic virotherapy, as given by the reaction–diffusion–taxis system \[\begin{eqnarray*} \left\{ \begin{array}{l} u_t = \Delta u - \nabla (u\nabla v)-\rho uz, \\ v_t = - (u+w)v, \\ w_t = D_w \Delta w - w + uz, \\ z_t = D_z \Delta z - z - uz + \beta w, \end{array} \right. \end{eqnarray*}\] in a smoothly bounded domain Ω ⊂ ℝ2, with parameters D w  > 0, D z  > 0, β > 0 and ρ ⩾ 0. Previous analysis has asserted that for all reasonably regular initial data, an associated no-flux type initial-boundary value problem admits a global classical solution, and that this solution is bounded if β < 1, whereas whenever β > 1 and $({1}/{|\Omega |})\int _\Omega u(\cdot ,0) > 1/(\beta -1)$ , infinite-time blow-up occurs at least in the particular case when ρ = 0. In order to provide an appropriate complement to this, the current study reveals that for any ρ ⩾ 0 and arbitrary β > 0, at each prescribed level γ ∈ (0, 1/(β − 1)+) one can identify an L∞-neighbourhood of the homogeneous distribution (u, v, w, z) ≡ (γ, 0, 0, 0) within which all initial data lead to globally bounded solutions that stabilize towards the constant equilibrium (u∞, 0, 0, 0) with some u∞ > 0.

A critical virus production rate for efficiency of oncolytic virotherapy

2020 ◽  
pp. 1-16 ◽  
Author(s):  
YOUSHAN TAO ◽  
MICHAEL WINKLER
Keyword(s):  
Initial Data ◽  
Virus Production ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Threshold Value ◽  
Oncolytic Virotherapy ◽  
Formula One ◽  
Image Position ◽  
Initial Boundary Value

In a planar smoothly bounded domain $\Omega$ , we consider the model for oncolytic virotherapy given by $$\left\{ \begin{array}{l} u_t = \Delta u - \nabla \cdot (u\nabla v) - uz, \\[1mm] v_t = - (u+w)v, \\[1mm] w_t = d_w \Delta w - w + uz, \\[1mm] z_t = d_z \Delta z - z - uz + \beta w, \end{array} \right.$$ with positive parameters $ D_w $ , $ D_z $ and $\beta$ . It is firstly shown that whenever $\beta \lt 1$ , for any choice of $M \gt 0$ , one can find initial data such that the solution of an associated no-flux initial-boundary value problem, well known to exist globally actually for any choice of $\beta \gt 0$ , satisfies $$u\ge M \qquad \mbox{in } \Omega\times (0,\infty).$$ If $\beta \gt 1$ , however, then for arbitrary initial data the corresponding is seen to have the property that $$\liminf_{t\to\infty} \inf_{x\in\Omega} u(x,t)\le \frac{1}{\beta-1}.$$ This may be interpreted as indicating that $\beta$ plays the role of a critical virus replication rate with regard to efficiency of the considered virotherapy, with corresponding threshold value given by $\beta = 1$ .

Estimating the Interaction Kernel in a Mathematical Model for a Beam With Internal Damping Mechanism

1995 ◽  
Author(s):  
Tao Lin ◽  
David L. Russell
Keyword(s):  
Mathematical Model ◽  
Fiber Composite ◽  
Mass Density ◽  
Integral Kernel ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Transverse Displacement ◽  
Internal Damping ◽  
Interaction Kernel ◽  
Initial Boundary Value

Abstract Beams formed by long fiber composite materials have certain internal damping torque. A mathematical model for the displacement of this type of beams in cantilever configuration is the following initial-boundary value problem of an integro-differential equation [1, 14]: (1) ρ ( x ) w t t ( x , t ) − 2 ( ∫ 0 L h ( x , y ) [ w t x ( x , t ) − w t x ( y , t ) ] d y ) x + ( E I w x x ( x , t ) ) x x = f ( x , t ) , (2) w ( 0 , t ) = 0 , w x ( 0 , t ) = 0 , (3) w x x ( L , t ) = b l 1 ( t ) , (4) − ( E I w x x ( x , t ) ) x | x = L + 2 ∫ 0 L h ( L , y ) [ w t x ( L , t ) − w t x ( y , t ) ] d y = b l 2 ( t ) , (5) w ( x , 0 ) = w 0 ( x ) , w t ( x , 0 ) = w 1 ( x ) , where L is length of the beam, w(x, t) is the transverse displacement of the beam at time t and position x, ρ(x) is the mass density, EI is the stiffness parameter. The interaction integral kernel h(x, ξ) is introduced in this model by considering a restoring torque which comes from spatially variable bending of the beam. This kernel h(x, ξ) depends on the material properties of the beam. Choosing a different material (different h(x, ξ)) can realize a different damping effect for the beam.

Piecewise continuous solutions of nonlinear pseudoparabolic equations in two space dimensions*

1992 ◽  
Vol 121 (3-4) ◽  
pp. 203-217 ◽  
Author(s):  
Dao-Qing Dai ◽  
Wei Lin
Keyword(s):  
Lipschitz Constant ◽  
Complex Function ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Lipschitz Continuous ◽  
Two Space Dimensions ◽  
Image Position ◽  
Initial Boundary Value ◽  
Pseudoparabolic Equations ◽  
Nonlinear Pseudoparabolic Equation

SynopsisAn initial boundary value problem of Riemann type is solved for the nonlinear pseudoparabolic equation with two space variablesThe complex functionHis measurable on ℂ ×I × ℂ5, withIbeing an interval of the real line ℝ, Lipschitz continuous with respect to the last five variables, with the Lipschitz constant for the last variable being strictly less than one (ellipticity condition). No smallness assumption is needed in the argument.

The wave front set of the solution of a simple initial-boundary value problem with glancing rays. II

1977 ◽  
Vol 81 (1) ◽  
pp. 97-120 ◽  
Author(s):  
F. G. Friedlander ◽  
R. B. Melrose
Keyword(s):  
Boundary Value Problem ◽  
Wave Front ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Principal Result ◽  
Wave Front Set ◽  
Precise Meaning ◽  
Image Position ◽  
Initial Boundary Value

This paper is a sequel to an earlier paper in these Proceedings by one of us ((5); this will be referred to as [I]). The question considered there was that of determining the wave front set of the solution of the boundary value problemwhere x∈+, y∈n, and n > 1; the precise meaning of the boundary condition at x = 0 is explained in section 1 below. The principal result of [I] can be expressed concisely by saying that singularities do not propagate along the boundary; a detailed statement is given in Theorem 1·9 of the present paper.

The Initial Boundary-Value Problem for a Mathematical Model for Granular Medium

2015 ◽  
Vol 725-726 ◽  
pp. 863-868
Author(s):  
Vladimir Lalin ◽  
Elizaveta Zdanchuk
Keyword(s):  
Mathematical Model ◽  
Boundary Value Problem ◽  
Stress Tensor ◽  
Granular Medium ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Cosserat Continuum ◽  
Suitable Model ◽  
Initial Boundary Value

In this work we consider a mathematical model for granular medium. Here we claim that Reduced Cosserat continuum is a suitable model to describe granular materials. Reduced Cosserat Continuum is an elastic medium, where all translations and rotations are independent. Moreover a force stress tensor is asymmetric and a couple stress tensor is equal to zero. Here we establish the variational (weak) form of an initial boundary-value problem for the reduced Cosserat continuum. We calculate the variation of corresponding Hamiltonian to obtain motion differential equation.

Boundedness and stabilization in a multi-dimensional chemotaxis—haptotaxis model

2014 ◽  
Vol 144 (5) ◽  
pp. 1067-1084 ◽  
Author(s):  
Youshan Tao ◽  
Michael Winkler
Keyword(s):  
Quantitative Result ◽  
Domain Of Attraction ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Flux Boundary Conditions ◽  
The Stability ◽  
Image Position ◽  
Initial Boundary Value ◽  
State 1 ◽  
Uniformly Bounded

This paper deals with the coupled chemotaxis-haptotaxis model of cancer invasion given bywhereχ, ξandμare positive parameters andΩ ⊂ ℝn(n≥ 1) is a bounded domain with smooth boundary. Under zero-flux boundary conditions, it is shown that, for anyμ>χand any sufficiently smooth initial data (u0,w0) satisfyingu0≥ 0 andw0> 0, the associated initial–boundary-value problem possesses a unique global smooth solution that is uniformly bounded. Moreover, we analyse the stability and attractivity properties of the non-trivial homogeneous equilibrium (u, v, w) ≡ (1,1, 0) and establish a quantitative result relating the domain of attraction of this steady state to the size ofμ. In particular, this will imply that wheneveru0> 0 and 0 <w0< 1 inthere exists a positive constantμ* depending only onχ, ξ, Ω, u0andw0such that for anyμ<μ* the above global solution (u, v, w) approaches the spatially uniform state (1, 1, 0) as time goes to infinity.

A model for the two-way propagation of water waves in a channel

1976 ◽  
Vol 79 (1) ◽  
pp. 167-182 ◽  
Author(s):  
Jerry L. Bona ◽  
Ronald Smith
Keyword(s):  
Water Waves ◽  
Open Channel ◽  
Boussinesq Equations ◽  
Coupled System ◽  
Finite Amplitude ◽  
Regularity Of Solutions ◽  
Initial Value ◽  
One Dimensional ◽  
Image Position ◽  
Continuous Dependence Of Solutions

Global existence, uniqueness and regularity of solutions and continuous dependence of solutions on varied initial data are established for the initial-value problem for the coupled system of equationsThis system has the same formal justification as a model for the two-way propagation of (one-dimensional) long waves of small but finite amplitude in an open channel of water of constant depth as other versions of the Boussinesq equations. A feature of the analysis is that bounds on the wave amplitude η are obtained which are valid for all time.

Landau–Ginzburg-type equations on the half-line in the critical case

2005 ◽  
Vol 135 (6) ◽  
pp. 1241-1262 ◽  
Author(s):  
Elena I. Kaikina ◽  
Hector F. Ruiz-Paredes
Keyword(s):  
Critical Case ◽  
Linear Part ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Inverse Laplace Transform ◽  
Half Line ◽  
Global Existence Of Solutions ◽  
Representation Of Solutions ◽  
Image Position ◽  
Initial Boundary Value

We study nonlinear Landau–Ginzburg-type equations on the half-line in the critical case where β ∈ C, ρ > 2. The linear operator K is a pseudodifferential operator defined by the inverse Laplace transform with dissipative symbol K(p) = αpρ, M = [1/2ρ]. The aim of this paper is to prove the global existence of solutions to the initial–boundary-value problem and to find the main term of the asymptotic representation of solutions in the critical case, when the time decay of the nonlinearity has the same rate as that of the linear part of the equation.

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