Via Carleman estimates we prove uniqueness and continuous dependence results for the temporal
traces of solutions to three overdetermined linear parabolic ill-posed problems with no initial condition, the first being integro-differential, the latter two being with deviating arguments. The overdetermination is prescribed in an open subset of the (space-time) lateral boundary.
We prove existence and uniqueness results
for mild solutions to direct and inverse Cauchy problems
related to linear first-order partial differential equations in Banach spaces. Some applications are given to ultraparabolic differential and integrodifferential problems.
In this paper we study two problems concerned with recovering memory kernels related to two sub-bodies Ω1 and Ω2 of an open thermal body under the assumptions that and is not accessible for the measurements. Additional measurements of temperature gradient or flux type are provided on ∂Ω. In the first problem the memory kernel related to Ω1 is unknown and a single measurement is given. In the second problem both kernels are to be determined from two measurements on ∂Ω. Making use of Laplace transforms, we prove the uniqueness for these identification problems in the infinite time interval (0, ∞).
We consider the following identification problems in a general Banach space X: find a function u : [0, T] → X and a vector z ∈ X such that the initial-value problems
are fulfilled, along with the nonlocal additional condition ∫[0,T]u(t)dμ(t) = φ ∈ X, for some probability Borel probability measure μ on the interval [0, T]. Here A : D(A) ⊂ X → X is a (possibly unbounded) closed linear operator, h, k and ƒ are scalar functions and g is a X-valued source term. We recall that the same problem with h = k = 0 has been previously studied by Anikonov and Lorenzi in [J. Inverse Ill-posed Probl. 7: 669–681, 2007], Prilepko, Piskarev and Shaw in [J. Inverse Ill-Posed Probl. 15: 831–851, 2007], and subsequently generalized by Lorenzi and Vrabie in [Discr. Continuous Dynam. Syst., 2011]. Under suitable assumptions on the structural data of the problem, we prove local-in-time existence and uniqueness for the function u, and an explicit representation formula for z depending on u. Also, a continuous dependence of Lipschitz type of the solution (u, z) on the data is provided. Finally, two applications to parabolic integro-differential boundary value problems are considered.
Via Carleman estimates we determine sufficient conditions ensuring uniqueness and continuous dependence results for a severely ill-posed linear integro-differential boundary-value parabolic problem with no initial condition. This latter condition is replaced with an additional boundary information prescribing the temperature on an open subset of the geometric domain .
The integral operators entering the equation are defined by integrals of Volterra type with respect to time.
The aim of the present paper is to generalize the results in [Denisov, J. Inverse Ill-Posed Probl. 16: 837–848, 2008.] devoted to a one-dimensional semilinear wave equation and consisting of recovering a time-dependent function α representing the transformed argument. More exactly: (i) we will deal with a general semilinear integro-differential hyperbolic d-dimensional equation in divergence form (d = 1, 2, 3); (ii) the space-time set considered here is a smooth cylinder where surface boundary conditions are prescribed; (iii) the term with transformed arguments is allowed to contain integral operators; (iv) the additional information is of integral type.
The existence and uniqueness results for our specific problem are deduced as consequences of similar results for an operator integro-differential identification problem in a Hilbert space.
We prove uniqueness and continuous dependence results for a severely ill-posed linear integrodifferential
boundary-value parabolic problem with no initial condition. This latter condition is replaced with an additional
boundary information prescribing the temperature on an open subset of the geometric domain .
The integral operators entering the equation are defined by integrals of Volterra type with respect to time. In particular, the class of integrodifferential equations dealt with in this paper include those occurring in the linear
theory of heat flow in a rigid body consisting of a material with thermal memory.