Global uniqueness and Hölder stability for recovering a nonlinear source term in a parabolic equation

Abstract

Consider the semilinear parabolic equation −u t (x, t) + u xx + q(u) = f (x, t), with the initial condition u(x, 0) = u 0 (x), Dirichlet boundary conditions u(0, t) = ϕ 0 (t), u(1, t) = ϕ 1 (t) and a sufficiently regular source term q(•), which is assumed to be known a priori on the range of u 0 (x). We investigate the inverse problem of determining the function q(•) outside this range from measurements of the Neumann boundary data u x (0, t) = ψ 0 (t), u x (1, t) = ψ 1 (t). Via the method of Carleman estimates, we derive global uniqueness of a solution (u, q) to this inverse problem and Hölder stability of the functions u and q with respect to errors in the Neumann data ψ 0 , ψ 1 , the initial condition u 0 and the a priori knowledge of the function q (on the range of u 0). The results are illustrated by numerical tests. The results of this paper can be extended to more general nonlinear parabolic equations.

Some inverse stability results for the bistable reaction–diffusion equation using Carleman inequalities

“…(i) We suppose that a satisfy (2) and α satisfy (3). Let v be the solution of (1) with initial condition v 0 ∈ L 2 (Ω) and let v be another solution of the same system with initial …”

“…a and a satisfy (2) and α and α satisfy(3). We denote by (p, p) = (a, a) or (α, α), and by v = v(v 0 , p) and v = v(v 0 , p) the corresponding solutions of (1).…”

Determination of the insolation function in the nonlinear Sellers climate model

“…Yet this question leads to two main difficulties. Even if one assumes the coalbedo is smooth, the question of unconditional global stability results for a nonlinear smooth term in a parabolic equation is not well-understood (see [19] for a partial answer). If one considers the Budyko model (the coalbedo is seen as a maximal monotone graph), there are well-posedness problems such as non-uniqueness of solutions [16][17][18].…”

Identifiability analysis of an epidemiological model in a structured population

“…Results on various classes of linear models with pointwise observation where obtained using Carleman estimates, for instance for the Schrödinger equation [18] or for a non-stationary particle transport equation (see [19] and references therein). In the nonlinear case, we only found results dealing with parabolic equations using Carleman estimates [20][21][22].…”

“…It follows that (Y (a) −Ȳ (a)) and M ∂ a y (t, a) are collinear, which yields(20). Moreover, Y (a) − M y (t, a) andȲ (a) − Mȳ(t, a) are also collinear and consequently(21) holds.In the second case, (16) yields βI(t − a)( X(a) − y(t, a)) = 0. It can be easily checked that when starting from a positive infected population at time zero, I remains positive on [0, T ], so X(a) = y(t, a) =X(a).…”