Unique solvability of the inverse problem of determination of the leading coefficient in a parabolic equation

We study existence and uniqueness of the solution for the inverse problem of determination of the unknown coefficient ϱ(t) multiplying u _t in a nondivergence parabolic equation. As additional information, the integral of the solution over the domain of space variables with some given weight function is specified. The coefficients of the equation depend both on time and on the space variables.     

Two inverse problems of finding a coefficient in a parabolic equation

For a parabolic equation, we consider inverse problems of reconstructing a coefficient that depends on the space variables alone. The first problem is to find a lower-order coefficient c(x) multiplying u(x, t), and the second problem is to find the coefficient a(x) multiplying Δu. As additional information, the integral of the solution with respect to time with some weight function is given. The coefficients of the equation depend both on time and on the space variables. We obtain sufficient conditions for the existence of generalized solutions of our problems; moreover, for the first problem, we also prove uniqueness and construct an iterative sequence that converges to the desired coefficient almost everywhere in the domain. We present examples of input data of these problems for which the assumptions of our theorems are necessarily true.     

A mixed problem for the Laplace operator in a domain with moderately close holes

We investigate the behavior of the solution of a mixed problem in a domain with two moderately close holes. We introduce a positive parameter ε and we define a perforated domain Ω_ε obtained by making two small perforations in an open set. Both the size and the distance of the cavities tend to 0 as ε → 0. For ε small, we denote by u_ε the solution of a mixed problem for the Laplace equation in Ω_ε. We describe what happens to u_ε as ε → 0 in terms of real analytic maps and we compute an asymptotic expansion.     

Corrigendum to “Discrepancy principle for the dynamical systems method” [Communications in Nonlinear Science and Numerical Simulation 10 (2005) 95–101]

Consider an operator equation B(u) − f = 0 in a real Hilbert space. Let us call this equation ill-posed if the operator B′(u) is not boundedly invertible, and well-posed otherwise. The dynamical systems method (DSM) for solving this equation consists of a construction of a Cauchy problem, which has the following properties: (1) it has a global solution for an arbitrary initial data, (2) this solution tends to a limit as time tends to infinity, (3) the limit is the minimal-norm solution to the equation B(u) = f. A global convergence theorem is proved for DSM for equation B(u) − f = 0 with monotone operators B.     

Nonlinear waves in friedman-robertson-walker space-times

In this paper we consider the initial value problem for the nonlinear wave equation □u = F(u, u′) in Friedman-Robertson-Walker space-time, □ being the D'Alambertian in local coordinates of space-time. We obtain decay estimates and show that the equation has global solutions for small initial data. We do it by reducing the problem to an initial value problem for the wave equation over hyperbolic space. As byproduct we derive decay and global existence for solutions of the wave equation over the hyperbolic space with small initial data. The same technique with some auxiliary lemmas similar to the ones proved in [6], [7] can be used to generalize the result to the case when F depends also on second derivatives of u in a certain way.     

Homogenization of a boundary-value problem with a nonlinear boundary condition in a thick junction of type 3:2:1

We consider a boundary-value problem for the Poisson equation in a thick junction Ω_ε, which is the union of a domain Ω₀ and a large number of ε-periodically situated thin curvilinear cylinders. The following nonlinear Robin boundary condition ∂_νu_ε + εκ(u_ε)=0 is given on the lateral surfaces of the thin cylinders. The asymptotic analysis of this problem is performed as ε → 0, i.e. when the number of the thin cylinders infinitely increases and their thickness tends to zero. We prove the convergence theorem and show that the nonlinear Robin boundary condition is transformed (as ε → 0) in the blow-up term of the corresponding ordinary differential equation in the region that is filled up by the thin cylinders in the limit passage. The convergence of the energy integral is proved as well. Using the method of matched asymptotic expansions, the approximation for the solution is constructed and the corresponding asymptotic error estimate in the Sobolev space H¹(Ω_ε) is proved. Copyright © 2007 John Wiley & Sons, Ltd.     

Compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity

We study the branch of semistable and unstable solutions (i.e., those whose Morse index is at most 1) of the Dirichlet boundary value problem ? Δu = λf(x)/(1 ? u)² on a bounded domain Ω ? ?^N, which models—among other things—a simple electrostatic microelectromechanical system (MEMS) device. We extend the results of 11 relating to the minimal branch, by obtaining compactness along unstable branches for 1 ≤ N ≤ 7 on any domain Ω and for a large class of “permittivity profiles” f. We also show the remarkable fact that powerlike profiles f(x) ? |x|^α can push back the critical dimension N = 7 of this problem by establishing compactness for the semistable branch on the unit ball, also for N ≥ 8 and as long as As a byproduct, we are able to follow the second branch of the bifurcation diagram and prove the existence of a second solution for λ in a natural range. In all these results, the conditions on the space dimension and on the power of the profile are essentially sharp. © 2007 Wiley Periodicals, Inc.     

Destruction of dissipative structures under random actions

We consider random-parameter chemical kinetic systems that are important in numerous physical, chemical, and biological applications. Random parameters describe the action of ambient medium fluctuations on the system. We estimate the probability that the system state remains in a given domain of the phase space during a time interval [0, T] under the condition that the state at the initial instant was in u ₀ , where u ₀ is the equilibrium solution describing a dissipative structure. We show that in some cases, the problem of maximizing this probability is reducible to the known problem of minimizing the Hopfield Hamiltonian.     

Remarks on Nonlinear Neumann Problems in Periodic Domains

We study a semilinear elliptic equation Au = f(x, u) with nonlinear Neumann boundary condition Bu = φ(ξ, u) in an unbounded domain Ω ? ?ⁿ, the boundary of which is defined by periodic functions. We assume that f and φ and the coefficients of the operators are asymptotically periodic in the space variables. Our main result states the existence of an asymptotically decaying, nontrivial solution of this problem with minimal energy. The proof is based on the concentration-compactness principle.     

Final State Problem for the Cubic Nonlinear Schrödinger Equation with Repulsive Delta Potential

We consider the asymptotic behavior in time of solutions to the cubic nonlinear Schrödinger equation with repulsive delta potential (δ-NLS). We shall prove that for a given small asymptotic profile u _ap , there exists a solution u to (δ-NLS) which converges to u _ap in L ²(?) as t → ∞. To show this result we exploit the distorted Fourier transform associated to the Schrödinger equation with delta potential.     

Stokes and Navier–Stokes equations with nonhomogeneous boundary conditions

In this paper, we study the existence and regularity of solutions to the Stokes and Oseen equations with nonhomogeneous Dirichlet boundary conditions with low regularity. We consider boundary conditions for which the normal component is not equal to zero. We rewrite the Stokes and the Oseen equations in the form of a system of two equations. The first one is an evolution equation satisfied by Pu, the projection of the solution on the Stokes space – the space of divergence free vector fields with a normal trace equal to zero – and the second one is a quasi-stationary elliptic equation satisfied by (I−P)u, the projection of the solution on the orthogonal complement of the Stokes space. We establish optimal regularity results for Pu and (I−P)u. We also study the existence of weak solutions to the three-dimensional instationary Navier–Stokes equations for more regular data, but without any smallness assumption on the initial and boundary conditions.     

Mixed Problem for an Ultraparabolic Equation in Unbounded Domain

We investigate a mixed problem for a nonlinear ultraparabolic equation in a certain domain Q unbounded in the space variables. This equation degenerates on a part of the lateral surface on which boundary conditions are given. We establish conditions for the existence and uniqueness of a solution of the mixed problem for the ultraparabolic equation; these conditions do not depend on the behavior of the solution at infinity. The problem is investigated in generalized Lebesgue spaces.     

Two-Scale Convergence Method and Scattering Problems

In this paper, we deal with the steady-state acoustic wave equation in the space ℝ³ diffracted by an obstacle made by an inhomogeneous medium and located in a bounded domain. The inhomogeneity of the medium depends on a parameter ε > 0. If the solution u _ε converges to a solution u ₀ of the limit problem as ε → 0, as in the homogenization process, then we can use the two-scale convergence method to study the convergence of the gradient.__________Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 10, Suzdal Conference-4, 2003.     

The Inverse Problem of Simultaneous Determination of the Two Time-Dependent Lower Coefficients in a Nondivergent Parabolic Equation in the Plane

We prove existence and uniqueness theorems for the solution of the inverse problem of simultaneous determination of the t-dependent coefficients of u and u_x in a nondivergent parabolic equation with two independent variables from integral observation of x. Estimates of the maxima of the moduli of these coefficients with constants explicitly expressed in terms of the input data of the problem are given. An example of an inverse problem to which the proved theorems apply is presented.

     

The boundary trace and generalized boundary value problem for semilinear elliptic equations with coercive absorption

In the first part of the paper we establish the existence of a boundary trace for positive solutions of the equation ?Δu + g(x, u) = 0 in a smooth domain Ω ? ?^N, for a general class of positive nonlinearities. This class includes every space independent, monotone increasing g which satisfies the Keller‐Osserman condition as well as degenerate nonlinearities g_α,q of the form g_α,q (x, u) = d(x, ?Ω)^α |u|^q?1 u, with α > ?2 and q > 1. The boundary trace is given by a positive regular Borel measure which may blow up on compact sets. In the second part we concentrate on the family of nonlinearities {g_α,q}, determine the critical value of the exponent q (for fixed α > ?2) and discuss (a) positive solutions with an isolated singularity, for subcritical nonlinearities and (b) the boundary value problem for ?Δu + g_α,q (x, u) = 0 with boundary data given by a positive regular Borel measure (possibly unbounded). We show that, in the subcritical case, the problem possesses a unique solution for every such measure. © 2003 Wiley Periodicals, Inc.     

Phragmen-Lindelof Principle for Some Quasilinear Evolution Equations of the Second Order

We consider the equation u _tt + A(u _t ) + B(u) = 0, where A and B are quasilinear operators with respect to the variable x of the second order and the fourth order, respectively. In a cylindrical domain unbounded with respect to the space variables, we obtain estimates that characterize the minimum growth of any nonzero solution of the mixed problem at infinity.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 2, pp. 239–249, February, 2005.     

Approximate solution of u′=–A2u using a relation between exp(–tA2) and exp(itA)

Let A be a positive definite operator in a Hilbert space and consider the initial value problem for u _t =–A² u. Using a representation of the semi group exp(–A² t) in terms of the group exp(iAt) we express u in terms of the solution of the standard heat equation w _t = w _yy , with initial values v solving the initial value problem for v _y = iAv. This representation is used to construct a method for approximating u in terms of approximations of v. In the case that A is a 2^nd order elliptic operator the method is combined with finite elements in the spatial variable and then reduces the solution of the 4th order equation for u to that of the 2^nd order equation for v, followed by the solution of the heat equation in one space variable.     

Boundary regularity for a family of overdetermined problems for the Helmholtz equation

If a nonconstant solution u of the Helmholtz equation exists on a bounded domain with u satisfying overdetermined boundary conditions (u and its normal derivative both required to be constant on the boundary), then under certain assumptions the boundary of the domain is proved to be real-analytic. Under weaker assumptions, if a real-analytic portion of the boundary has a real-analytic extension, then that extension must also be part of the boundary. Also, an explicit formula for u is given and a condition (which does not involve u) is given for a bounded domain to have such a solution u defined on it. Both of these last results involve acoustic single- and double-layer potentials.     

Thermal avalanche for blowup solutions of semilinear heat equations

We consider the semilinear heat equation u_t = Δu + u^p both in ?^N and in a bounded domain with homogeneous Dirichlet boundary conditions, with 1 < p < p_s where p_s is the Sobolev exponent. This problem has solutions with finite‐time blowup; that is, for large enough initial data there exists T < ∞ such that u is a classical solution for 0 < t < T, while it becomes unbounded as t ↗ T. In order to understand the situation for t > T, we consider a natural approximation by reaction problems with global solution and pass to the limit. As is well‐known, the limit solution undergoes complete blowup: after it blows up at t = T, the continuation is identically infinite for all t > T. We perform here a deeper analysis of how complete blowup occurs. Actually, the singularity set of a solution that blows up as t ↗ T propagates instantaneously at time t = T to cover the whole space, producing a catastrophic discontinuity between t = T^? and t = T⁺. This is called the “avalanche.” We describe its formation as a layer appearing in the limit of the natural approximate problems. After a suitable scaling, the initial structure of the layer is given by the solution of a limit problem, described by a simple ordinary differential equation. As t proceeds past T, the solutions of the approximate problems have a traveling wave behavior with a speed that we compute. The situation in the inner region depends on the type of approximation: a conical pattern is formed with time when we approximate the power u^p by a flat truncation at level n, while for tangent truncations we get an exponential increase in time and a diffusive spatial pattern. © 2003 Wiley Periodicals, Inc.     

On wellposedness in nonlinear acoustics

Motivated by a medical application from lithotripsy, we study the initial–boundary value problem given by Westervelt equation (1) in a bounded domain Ω. This models the nonlinear evolution of the acoustic pressure u excited at a part Γ₀ of the boundary. Along with the excitation given by Neumann boundary condition as in (1) , we also consider the Dirichlet type of excitation. Whereas shock waves are known to emerge after a sufficiently large time interval for appropriate initial and boundary conditions, we here prove existence and uniqueness as well as stability of a solution u for small data g, u₀ and u₁ or short time T, using a fixed point argument. Moreover we extend the result to the more general model given by the Kuznetsov equation (2) for the acoustic velocity potential ψ. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)