Structured Inverse Modeling in Parabolic Diffusion Problems

Inverse problems for parabolic equations

Let ut −∇2u = f(x) := ∑M m=1 amδ(x− xm) in D × [0,∞), where D ⊂ R3 is a bounded domain with a smooth connected boundary S, am = const, δ(x− xm) is the delta-function. Assume that u(x, 0) = 0, u = 0 on S. Given the extra data u(yk, t) := bk(t), 1 ≤ k ≤ K, can one find M,am, and xm? Here K is some number. An answer to this question and a method for finding M,am, and xm are given.

Structured Inverse Eigenvalue Problems

An inverse eigenvalue problem concerns the reconstruction of a structured matrix from prescribed spectral data. Such an inverse problem arises in many applications where parameters of a certain physical system are to be determined from the knowledge or expectation of its dynamical behavior. Spectral information is entailed because the dynamical behavior often is governed by the underlying natur...

Diffusion versus absorption in semilinear parabolic problems

We study the limit, when k → ∞, of the solutions u = uk of (E) ∂tu−∆u+ h(t)uq = 0 in RN × (0,∞), uk(., 0) = kδ0, with q > 1, h(t) > 0. If h(t) = e−ω(t)/t where ω > 0 satisfies to R 1 0 p ω(t)t−1dt < ∞, the limit function u∞ is a solution of (E) with a single singularity at (0, 0), while if ω(t) ≡ 1, u∞ is the maximal solution of (E). We examine similar questions for equations such as ∂tu−∆u + h...

Parameter determination in a parabolic inverse problem in general dimensions

It is well known that the parabolic partial differential equations in two or more space dimensions with overspecified boundary data, feature in the mathematical modeling of many phenomena. In this article, an inverse problem of determining an unknown time-dependent source term of a parabolic equation in general dimensions is considered. Employing some transformations, we change the inverse prob...

Variational Modeling of Parabolic Free Boundary Problems