Two-dimensional inverse quasilinear parabolic problem with periodic boundary condition

In this study, we consider a coefficient problem of a quasi-linear two-dimensional parabolic inverse problem with periodic boundary and integral over determination conditions. We prove the existence, uniqueness and continuously dependence upon the data of the solution by iteration method. Also, we consider numerical solution for this inverse problem by using linearization and the implicit finite-difference scheme.     

非线性抛物方程在非线性边界条件下解的爆破时间的下界

研究了非线性抛物方程在非线性边界条件下的解的爆破问题,通过构造一个能量表达式,运用微分不等式的方法,得到该能量表达式所满足的微分不等式,然后通过积分得到当爆破发生时解在非线性边界条件下的爆破时间的下界.     

Recovering unknown terms in a nonlinear boundary condition for Laplace's equation

^** Email: gabriele{at}fi.iac.cnr.it We consider a thin metallic plate whose top side is inaccessibleand in contact with a corroding fluid. Heat exchange betweenmetal and fluid follows linear Newton's cooling law as longas the inaccessible side is not damaged. We assume that theeffects of corrosion are modelled by means of a nonlinear perturbationin the exchange law. On the other hand, we are able to heatthe conductor and take temperature maps of the accessible side.Our goal is to recover the nonlinear perturbation of the exchangelaw on the top side from thermal data collected on the oppositeone (thermal imaging). In this paper, we use a stationary model,i.e. the temperature inside the plate is assumed to fulfil Laplace'sequation. Hence, our problem is stated as an inverse ill-posedproblem for Laplace's equation with nonlinear boundary conditions.We study identifiability and local Lipschitz stability. In particular,we prove that the nonlinear term is identified by one Cauchydata set. Moreover, we produce approximated solutions by meansof an optimizational method.     

Global existence of solutions of initial boundary value problem for nonlocal parabolic equation with nonlocal boundary condition

We prove the global existence and blow-up of solutions of an initial boundary value problem for nonlinear nonlocal parabolic equation with nonlinear nonlocal boundary condition. Obtained results depend on the behavior of variable coefficients for large values of time.     

The thermostat problem with a nonlocal nonlinear boundary condition

The thermostat controller for an air-conditioning system isusually placed in a position at some distance from the unitand this can lead to large swings in temperature. This paperaddresses this question by studying a paradigm—a one-dimensionalheat conduction equation with and without heat loss, and wherethe flux of heat extracted or input by the unit is consideredto be a function of the temperature at the other end. The essential results are that the system can be unstable andthat this is exacerbated both by a more powerful air-conditioningunit and by more efficient insulation.     

Inverse spectral problem for integro-differential operators

In this paper, we study the inverse spectral problem on a finite interval for the integro-differential operator ? which is the perturbation of the Sturm-Liouville operator by the Volterra integral operator. The potential q belongs to L ₂[0, π] and the kernel of the integral perturbation is integrable in its domain of definition. We obtain a local solution of the inverse reconstruction problem for the potential q, given the kernel of the integral perturbation, and prove the stability of this solution. For the spectral data we take the spectra of two operators given by the expression for ? and by two pairs of boundary conditions coinciding at one of the finite points.     

An inverse coefficient problem for a quasilinear parabolic equation with periodic boundary and integral overdetermination condition

In this paper, the inverse problem of finding the time‐dependent coefficient of heat capacity together with the solution periodic boundary and integral overdetermination conditions is considered. Under some natural regularity and consistency conditions on the input data, the existence, uniqueness, and continuous dependence upon the data of the solution are shown. Some considerations on the numerical solution for this inverse problem are presented with an example. Copyright © 2014 John Wiley & Sons, Ltd.     

Pseudo almost periodic solutions to parabolic boundary value inverse problems

We first define the pseudo almost periodic functions in a more general setting.Then we show the existence,uniqueness and stability of pseudo almost periodic solutions of parabolic inverse problems for a type of boundary value problems.     

Solvability for a nonlinear second-order three-point boundary value problem

This paper investigates the existence of nontrivial solution for the three-point boundary value problem

     

Minimum-time problem of a retarded parabolic system with the Neumann boundary condition

^** Email: skrakowi{at}usk.pk.edu.pl In this paper, the minimum-time problem for parabolic systemsin which retarded arguments appear in the integral form in thestate equation is presented. The particular properties of theoptimal control are proved.     

On the inverse problem of determining the right-hand side of a parabolic equation under an integral overdetermination condition

We study the unique solvability of the inverse problem of determining the righthand side of a parabolic equation whose leading coefficient depends on both the time and the spatial variable under an integral overdetermination condition with respect to time. We obtain two types of condition sufficient for the local solvability of the inverse problem as well as study the so-called Fredholm solvability of the inverse problem under consideration.Translated from Matematicheskie Zametki, vol. 77, no. 4, 2005, pp. 522–534.Original Russian Text Copyright © 2005 by V. L. Kamynin.This revised version was published online in April 2005 with a corrected issue number.     

Determination of an unknown function in a parabolic inverse problem by Sinc‐collocation method

In this article, an inverse problem of determining an unknown time‐dependent source term of a parabolic equation is considered. We change the inverse problem to a Volterra integral equation of convolution‐type. By using Sinc‐collocation method, the resulting integral equation is replaced by a system of linear algebraic equations. The convergence analysis is included, and it is shown that the error in the approximate solution is bounded in the infinity norm by the condition number and the norm of the inverse of the coefficient matrix multiplied by a factor that decays exponentially with the size of the system. Some examples are given to demonstrate the computational efficiency of the method. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1584–1598, 2010     

An inverse coefficient problem for a parabolic equation in the case of nonlocal boundary and overdetermination conditions

In this paper, the inverse problem of finding the time‐dependent coefficient of heat capacity together with the solution of heat equation with nonlocal boundary and overdetermination conditions is considered. The existence, uniqueness and continuous dependence upon the data are studied. Some considerations on the numerical solution for this inverse problem are presented with the examples. Copyright © 2010 John Wiley & Sons, Ltd.     

Asymptotic simplification for a reaction-diffusion problem with a nonlinear boundary condition

We study non-negative solutions of the porous medium equationwith a source and a nonlinear flux boundary condition, u_t =(u^m)_xx + u^p in (0, ), x (0, T); – (u^m)_x (0, t) = u^q (0,t) for t (0, T); u (x, 0) = u₀ (x) in (0, ), where m > 1,p, q > 0 are parameters. For every fixed m we prove thatthere are two critical curves in the (p, q-plane: (i) the criticalexistence curve, separating the region where every solutionis global from the region where there exist blowing-up solutions,and (ii) the Fujita curve, separating a region of parametersin which all solutions blow up from a region where both globalin time solutions and blowing-up solutions exist. In the caseof blow up we find the blow-up rates, the blow-up sets and theblow-up profiles, showing that there is a phenomenon of asymptoticsimplification. If 2q < p + m the asymptotics are governedby the source term. On the other hand, if 2q > p + m theevolution close to blow up is ruled by the boundary flux. If2q = p + m both terms are of the same order.     

Multiple solutions of a quasilinear elliptic problem involving nonlinear boundary condition on exterior domain

In this paper, we study the multiplicity of non‐negative solutions for the quasilinear p‐Laplacian equation with the nonlinear boundary condition (1) where Δ_p denotes the p‐Laplacian operator, defined by △ _pu = div( | ? u | ^{p ? 2} ? u),1 < p < N, Ω is a smooth exterior domain in . is the outward normal derivative, . The parameters p,q,r are either or . The weight functions a(x),h(x),g(x) satisfy some suitable conditions. Using the decomposition of the Nehari manifold and the variational methods, we prove that problem (1) has at least two positive solutions provided 0 < | λ | < λ₁ for some λ₁. Copyright © 2012 John Wiley & Sons, Ltd.     

A numerical study of parabolic problems with nonlinear boundary conditions

In this paper, we consider an initial‐boundary value problem for a parabolic equation with nonlinear boundary conditions. The solution to the problem can be expressed as a convolution integral of a Green's function and two unknown functions. We change the problem to a system of two nonlinear Volterra integral equations of convolution type. By using an explicit procedure on the basis of Sinc‐function properties, the resulting integral equations are replaced by a system of nonlinear algebraic equations, whose solution yields an accurate approximate solution to the parabolic problem. Some examples are considered to illustrate the ability of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.     

Global blow-up for a localized nonlinear parabolic equation with a nonlocal boundary condition

This paper deals with the blow-up properties of positive solutions to a nonlinear parabolic equation with a localized reaction source and a nonlocal boundary condition. Under certain conditions, the blowup criteria is established. Furthermore, when f(u)=up, 0<p?1, the global blowup behavior is shown, and the blowup rate estimates are also obtained.