Solvability Of Class Research Articles

Classical solvability to the free boundary problem for a foam drainage equation. II. From the interface to the bottom

We study a nonlinear partial differential equation describing the evolution of a foam drainage in one dimensional case which was proposed by Goldfarb et al. in 1988 in order to investigate the flow of liquid through channels (Plateau borders) and nodes (intersections of four channels) between the bubbles, driven by gravity and capillarity. Its mathematical studies so far are mainly restricted within numerical and particular solutions; as for mathematical analysis of it there are only a few. We prove that the free boundary problem for the foam drainage equation in the region below an interface to the bottom in a foam column admits a unique global-in-time classical solution by the standard classical mathematical method, the maximum principle. Moreover, the existence of its steady solution and its stability are shown.

Classical solvability to the free boundary problem for a foam drainage equation. I. From the top to the interface

We study a nonlinear partial differential equation describing the evolution of a foam drainage in one dimensional case which was proposed by Goldfarb et al. [Izv. Akad. Nauk SSSR Mekh. Ghidk. Gaza 2, 103–108 (1988)] in order to investigate the flow of liquid through channels (Plateau borders) and nodes (intersections of four channels) between the bubbles, driven by gravity and capillarity. Its mathematical studies so far are mainly restricted within numerical and particular solutions; as for mathematical analysis of it there are only a few. We prove that the free boundary problem for the foam drainage equation in the region from the top to the interface in a foam column admits a unique global-in-time classical solution by the standard classical mathematical method, the maximum principle and the comparison theorem. Moreover, the existence of its steady solution and its stability are discussed.

The classical solvability for a one-dimensional nonlinear thermoelasticity system with the far field degeneracy

Abstract We study the local classical solvability of the Cauchy problem to the equations of one-dimensional nonlinear thermoelasticity. The governing model is a coupled system of a nonlinear hyperbolic equation for the displacement and a parabolic equation for the temperature of the elastic material. We allow the hyperbolic equation to degenerate at spacial infinity, which results that the coefficients of the coupled system are not uniformly bounded and then the previous methods for the strictly hyperbolic-parabolic coupled systems are invalid. We introduce a suitable weighted norm to establish the local existence and uniqueness of classical solutions by the contraction mapping principle. The existence time T of the solution is independent of the spatial variable.

Global classical solvability and stabilization in a two-dimensional chemotaxis-Navier-Stokes system involving Dirichlet boundary conditions for the signal

The chemotaxis-Navier-Stokes system{nt+u⋅∇n=Δn−∇⋅(nχ(n)∇c),ct+u⋅∇c=Δc−nc,ut+(u⋅∇)u=Δu+∇P+n∇ϕ,∇⋅u=0 is considered in a smoothly bounded domain Ω⊂R2 under the boundary conditions(∇n−nχ(n)∇c)⋅ν=0,c=c⋆,u=0,x∈∂Ω,t>0 with a given nonnegative constant c⋆. It is shown that if χ∈C2([0,∞)) and χ(n)→0 as n→∞, then for all suitably regular initial data, an associated initial value problem possesses a globally defined and bounded classical solution. When ‖n0‖L1(Ω) and ‖c0‖L∞(Ω) are suitably small and c⋆≡0, we further obtain the stabilization of the classical solution.

On subsolutions and concavity for fully nonlinear elliptic equations

Abstract Subsolutions and concavity play critical roles in classical solvability, especially a priori estimates, of fully nonlinear elliptic equations. Our first primary goal in this paper is to explore the possibility to weaken the concavity condition. The second is to clarify relations between weak notions of subsolution introduced by Székelyhidi and the author, respectively, in attempt to treat equations on closed manifolds. More precisely, we show that these weak notions of subsolutions are equivalent for equations defined on convex cones of type 1 in the sense defined by Caffarelli, Nirenberg and Spruck.

Nachal'no-kraevye zadachi dlya odnorodnykh parabolicheskikh sistem v poluogranichennoy ploskoy oblasti i uslovie dopolnitel'nosti

We consider initial–boundary value problems for homogeneous parabolic systemswith coefficients satisfying the double Dini condition with zero initial conditions in a semiboundedplane domain with nonsmooth lateral boundary. The method of boundary integralequations is used to prove a theorem on the unique classical solvability of such problems in thespace of functions that are continuous together with their first spatial derivative in the closureof the domain. An integral representation of the obtained solutions is given. It is shown thatthe condition for the solvability of the posed problems considered in the paper is equivalent tothe well-known complementarity condition.

Diffusion in media with membranes and some nonlocal parabolic problems

UDC 519.21 We establish the classical solvability of a certain conjugation problem for one-dimensional (with respect to a spatial variable) Kolmogorov backward equation with discontinuous coefficients and some variants of the general nonlocal Feller–Wentzell boundary condition given on nonsmooth boundaries of considered curvilinear domains. In addition, we prove, that the two-parameter Feller semigroup defined by the solution of this problem describes some inhomogeneous diffusion process with moving membranes on the given region of the real line. We also show the relationship between the constructed process and the generalized diffusion in the sense of M. I. Portenko.

Global classical solvability and asymptotic behaviors of a parabolic-elliptic Chemotaxis-type system modeling crime activities

This paper is devoted to a two-dimensional parabolic-elliptic system in crime activities. The main difference between this system and the chemotaxis system with logarithmic singular sensitivity and quadratic logistic source is that the source term of this system is a mixtype quadratic term. The local existence of solutions to its intial boundary valve problem associated with homogeneous Neumann boundary conditions was considered by Rodríguez. The present study indicates that for any properly regular initial data there admits global classical solutions. Moreover, the qualitative behaviors of solutions in large time scales are considered as well.

NONLINEAR INVERSE BOUNDARY VALUE PROBLEM FOR A SECOND ORDER PARABOLIC EQUATION WITH NONLOCAL BOUNDARY CONDITIONS

The paper studies the classical solvability of an inverse boundary value problem for a second order parabolic equation with nonlocal boundary conditions. For this purpose, first, the considered problem is reduced to an auxiliary equivalent problem in a certain sense. Then, using the Fourier method the auxiliary problem is presented as a system of integral equations. Further, by means of the contraction mappings principle the unique existence of the solution of the obtained system of integral equations is shown. At the end of investigation the existence and uniqueness theorem for the classical solution of the original inverse boundary value problem is proved based on the equivalence of these problems

Global classical solvability and stabilization in a two-dimensional chemotaxis–fluid system with sub-logarithmic sensitivity

In this paper, we consider the following system: [Formula: see text] in a smoothly bounded domain [Formula: see text] with [Formula: see text] and a given function [Formula: see text] with [Formula: see text] It is proved that if [Formula: see text] then for appropriately small initial data an associated no-flux/no-flux/Dirichlet initial-boundary value problem is globally solvable in the classical sense, and that if [Formula: see text] then under a different but still suitable smallness restriction of the initial data, a corresponding initial-boundary value problem subject to no-flux/no-flux/Dirichlet boundary conditions admits a unique classical solution which is globally bounded and approaches a constant equilibria [Formula: see text] in [Formula: see text] as [Formula: see text] with [Formula: see text]

Chemotaxis-driven stationary and oscillatory patterns in a diffusive HIV-1 model with CTL immune response and general sensitivity.

In this paper, a reaction-diffusion-chemotaxis HIV-1 model with a cytotoxic T lymphocyte (CTL) immune response and general sensitivity is investigated. We first prove the global classical solvability and L∞-boundedness for the considered model in a bounded domain with arbitrary spatial dimensions, which extends the previous existing results. Then, we apply the global existence result to the case with a linear proliferation immune response and an incidence rate. We study the spatiotemporal dynamics about the three types of spatially homogeneous steady states: infection-free steady state S0, CTL-inactivated infection steady state S1, and CTL-activated infection steady state S∗. Our analyses indicate that S0 is globally asymptotically stable if the basic reproduction number R0 is less than 1; if R0 is between 1 and a threshold, then S1 is globally asymptotically stable. However, if R0 is larger than the threshold, then the chemoattraction and chemorepulsion can destabilize S∗, and thus, a spatiotemporal pattern forms as the chemotactic sensitivity crosses certain critical values. We obtain two kinds of important patterns, which are induced by chemotaxis: stationary Turing pattern and irregular oscillatory pattern. We also find that different chemotactic response functions can affect system's dynamics. Based on some empirical parameter values, numerical simulations are given to illustrate the effectiveness of the theoretical predications.

Solutions with exponential singularity for (3 + 1)-D Protter problems

In the 1950s, Protter proposed multi-dimensional analogues of the classical Guderley–Morawetz problem for mixed-type hyperbolic-elliptic equations on the plane that models transonic flows in fluid dynamics. The multi-dimensional variants turn out to be different from the two-dimensional case and the situation there is still not clear. Here, we study Protter problems in the hyperbolic part of the domain. Unlike the planar analogues, the four-dimensional variant is not well-posed for classical solutions. The problem is not Fredholm — there is an infinite number of necessary conditions for classical solvability. Alternatively, the notion of a generalized solution that may have singularities was introduced. It is known that for smooth right-hand sides, the uniquely determined generalized solution may have a power-type growth at one boundary point. The singularity is isolated at the vertex of the boundary characteristic light cone and does not propagate along the cone. Here, we construct a new singular solution with an exponential growth at the point where the singularity appears.

Cauchy–Dirichlet Problem to Semilinear Multi-Term Fractional Differential Equations

In this paper, we analyze the well-posedness of the Cauchy–Dirichlet problem to an integro-differential equation on a multidimensional domain Ω⊂Rn in the unknown u=u(x,t), Dtν0(ϱ0u)−Dtν1(ϱ1u)−L1u−∫0tK(t−s)L2u(x,s)ds=f(x,t)+g(u),0<ν1<ν0<1, where Dtνi are the Caputo fractional derivatives, ϱi=ϱi(x,t) with ϱ0≥μ0>0, and Li are uniform elliptic operators with time-dependent smooth coefficients. The principal feature of this equation is related to the integro-differential operator Dtν0(ϱ0u)−Dtν1(ϱ1u), which (under certain assumption on the coefficients) can be rewritten in the form of a generalized fractional derivative with a non-positive kernel. A particular case of this equation describes oxygen delivery through capillaries to tissue. First, under proper requirements on the given data in the linear model and certain relations between ν0 and ν1, we derive a priori estimates of a solution in Sobolev–Slobodeckii spaces that gives rise to providing the Hölder regularity of the solution. Exploiting these estimates and constructing appropriate approximate solutions, we prove the global strong solvability to the corresponding linear initial-boundary value problem. Finally, obtaining a priori estimates in the fractional Hölder classes and assuming additional conditions on the coefficients ϱ0 and ϱ1 and the nonlinearity g(u), the global one-valued classical solvability to the nonlinear model is claimed with the continuation argument method.

Global existence and stabilization in a two-dimensional chemotaxis-Navier-Stokes system with consumption and production of chemosignals

We are concerned with the initial boundary value problems of chemotaxis-Navier-Stokes systems involving double chemosignals: one is an attractant consumed by the cells themselves and the other is an attractant or a repellent produced by the cells themselves. This work reveals that for any properly regular initial data, if the initial total cells mass is less than some threshold, then the corresponding attraction-atraction Navier-Stokes system admits a unique global classical solution, which, under extra two smallness assumptions on the initial data, is globally bounded and converges at exponential rate to the spatial equilibrium in the large time limit. As byproducts, the global classical solvability and stabilization of the Keller-Segel Navier-Stokes system and the uniform boundedness and large time behavior of the global classical solution of the associated attraction-repulsion Navier-Stokes system are simultaneously achieved.

Stationary and oscillatory patterns of a food chain model with diffusion and predator‐taxis

In this paper, we investigate pattern dynamics in a reaction‐diffusion‐chemotaxis food chain model with predator‐taxis, which extends previous studies of reaction‐diffusion food chain model. By virtue of diffusion semigroup theory, we first prove global classical solvability and boundedness for the considered model over a bounded domain with smooth boundary for arbitrary predator‐taxis sensitivity coefficient. Then the linear stability analysis for the considered model shows that chemotaxis can induce the losing of stability of the unique positive spatially homogeneous steady state via Turing bifurcation and Turing‐spatiotemporal Hopf bifurcation. These bifurcations results in the formation of two kinds of important spatiotemporal patterns: stationary Turing pattern and oscillatory pattern. Simultaneously, the threshold values for Turing bifurcation and Turing‐spatiotemporal Hopf bifurcation are given explicitly. Finally, numerical simulations are performed to illustrate and support our theoretical findings, and some interesting non‐Turing patterns are found in temporal Hopf parameter space by numerical simulation.

A note on second derivative estimates for Monge-Ampère-type equations

Abstract In this article, we revisit previous Pogorelov-type interior and global second derivative estimates of N. S. Trudinger, F. Jiang, and J. Liu for solutions of Monge-Ampère-type partial differential equations. Taking account of recent strict convexity regularity results of Guillen-Kitagawa and Rankin and following our earlier work in the optimal transportation case, we remove the monotonicity assumptions in the more general case of generated Jacobian equations and consequently in the subsequent application to classical solvability and global regularity for second boundary value problems.

Analysis of a chemotaxis-SIS epidemic model with unbounded infection force

A susceptible–infected–susceptible (SIS) model is considered which includes both taxis type cross-diffusive motion of susceptibles and a mass action infection mechanism. An associated no-flux initial–boundary value problem in one-dimensional bounded intervals is shown to admit globally bounded classical solutions for initial data of arbitrary size. For a higher-dimensional counterpart, a result on global classical solvability in the presence of suitably small initial data is derived. This is complemented by a statement on large time stabilization of small-data trajectories toward spatially constant and infection-free equilibria under an assumption on strict positivity of the considered space-dependent recovery rate.

INVERSE PROBLEM ON DETERMINING MANY UNKNOWNS FROM SCHWARTZ-TYPE DISTRIBUTIONS

We find the sufficient conditions for the unique (local in time) solvability of an inverse problem of finding m unknown functions $R_l(x)$, $l\in \{1,\dots,m\}$ from the Schwartz-type distributions $S'_{\gamma,(a)}(\Bbb R^n)$ in a source term of a diffusion equation \[D^{\beta}_t u-A(x,D)u=\sum\limits_{l=1}^mR_l(x)g_l(t)+F(x,t), \;\; (x,t) \in Q=\Bbb R^n\times (0,T] \] with the Djrbasian-Nersesian-Caputo time-fractional derivative of the order $\beta\in (0,1)$ where $A(x,D)$ is an elliptic differential operator of the second order, \[S_{\gamma,(a)}(\Bbb R^n)=\{v\in C^\infty(\Bbb R^n): ||v||_{k,(a)}=\sup\limits_{|\alpha|\le k,x\in \Bbb R^n}e^{a(1-\frac{1}{k}) |x|^{\frac{1}{\gamma}}}|D^{\alpha}v(x)|<+\infty\;\;\forall k\in \Bbb N, k\ge 2\}.\] We use time-integral over-determination conditions \[\frac{1}{T}\int_{0}^{T}u(x,t)\eta_l(t)dt=\Phi_l(x), \;\;x\in \Bbb R^n, \;\;l\in \{1,\dots,m\}\] with the given $\eta_l\in C^1[0,T]$ and Schwartz-type distributions $\Phi_l(x)$, $l\in \{1,\dots,m\}$. Note that time-integral over-determination conditions were used in the study of various inverse problems in various functional spaces. By properties of the Green vector-function the problem boils down to solving linear operator equation of the second kind with respect to the unknown solution $u$ of the Cauchy problem, continuous with values in Schwartz-type distributions, and a linear inhomogeneous algebraic system of equations for finding expressions of unknown functions $R_l(x)$, $l\in \{1,\dots,m\}$ through it. We generalize the results of [11] on the classical solvability of a problem with two unknown functions from Schwartz-type spaces of rapidly decreasing functions at infinity on the right-hand side of such an equation.

Global solutions to a Keller-Segel-consumption system involving singularly signal-dependent motilities in domains of arbitrary dimension

This manuscript considers the no-flux initial-boundary problem for the migration-consumption system{ut=Δ(uϕ(v)),vt=Δv−uv,(⋆) in a smoothly bounded domain Ω⊂Rn, n≥1, where ϕ suitably generalizes the singular prototype given byϕ(ξ)=ξ−α,ξ>0, with α>0.It is firstly shown that for such diffusion singularities of arbitrary strength, and for any given initial data u0 and v0 from W1,∞(Ω) with u0≥0 and v0>0 in Ω‾, a so-called very weak-strong solution (u,v) with (u,v)|t=0=(u0,v0) can be constructed. Under the additional restrictions that 2≤n≤5 and α>n−26−n, and under an additional assumption on ϕ′, it is furthermore asserted that the flux ∇(uϕ(v)) belongs to some reflexive Lebesgue space, and that (⋆) is satisfied in a standard weak sense.Finally, in the one-dimensional case a statement on global classical solvability is derived under the mere condition that ϕ∈C3((0,∞)) be positive.

Global classical solvability in a food metric chemotaxis system under zero boundary conditions at infinity

We consider the nutrient–chemotaxis model, derived by a food metric, on the real line. The model consists of the equation for the nutrient density of ordinary differential equation type and that for the microorganism density of parabolic type, in which the diffusion coefficient is singular at the zero nutrient density. We establish a global existence result of bounded classical solutions for a large class of initial data under zero boundary conditions at infinity. To this end, we propose a transformation related to the food metric that transforms the original system into a parabolic–hyperbolic system with linear diffusion. We obtain a proper global well-posedness result for the transformed system using standard estimates, including the heat kernel estimates. We convert this to the solution of the original equation via an inverse transformation.