An inverse source problem for generalized Rayleigh-Stokes equations involving superlinear perturbations

Abstract

We deal with the problem of identifying a source term in the Rayleigh-Stokes type equation with a nonlinear perturbation, where the nonlinearity may have a superlinear growth and the additional measurement is given at final time and depends on the state. Our aim is to prove the unique solvability and stability of solution. Furthermore, we show that the obtained solution is differentiable and it is the strong one.

Introduction

Let $\mathrm{\Omega }\subset {\mathbb{R}}^d$ be a bounded domain with smooth boundary ∂Ω. Consider the following problem: find $(f,u)$ satisfying

$${\partial }_{t}u-(1+D_t^{\{m\}})\mathrm{\Delta }u=f(x)g(t)+h(u)\text{ in }\mathrm{\Omega },t\in (0,T),$$

$$u=0\text{ on }\partial \mathrm{\Omega },t\ge 0,$$

$$u(0)=\xi \text{ in }\mathrm{\Omega },$$

$$u(T)=\phi (u)\text{ in }\mathrm{\Omega },$$

where $u(t)$ takes values in $L^2(\mathrm{\Omega })$, g, h and φ are given functions, ${\partial }_t=\frac{\partial }{\partial t}$, $D_t^{\{m\}}$ stands for the nonlocal differential operator of Riemann-Liouville type defined by

$$D_t^{\{m\}}v(t)=\frac{d}{dt}\underset{0}{\overset{t}{\int }}m(t-s)v(s)ds,$$

with respect to the kernel function $m\in L_{loc}^1({\mathbb{R}}^{+})$.

The aforementioned system is an inverse problem in the situation that the source term $F(x,t)=f(x)g(t)$ is not fully obtained from observations. In addition, in practical systems, a nonlinear perturbation may involve, which appears as function h. In order to identify f, we usually need a measurement at final time, e.g. condition (1.4). This condition is imposed in the case the measured data is possibly implicit, i.e., it depends on the mass/energy of the system. Concerning the structure of the source function in application, $f(x)$ can be interpreted as the spatial distribution of source/force which is supposed to be unknown, meanwhile $g(t)$ stands for the intensity depending on the time.

Regarding our model, (1.1) is a generalization for a number of problems considered in literature. In the case m is a constant (such that $1+m>0$), (1.1) is classical diffusion equation. If m is a regular function, e.g. $m\in C^1({\mathbb{R}}^{+})$ then our equation reads

$${\partial }_{t}u-(1+m_0)\mathrm{\Delta }u-\underset{0}{\overset{t}{\int }}{m}_1(t-s)\mathrm{\Delta }u(s)ds=f(x)g(t)+h(u),$$

with $m_0=m(0)$ and $m_1(t)=m^{\prime }(t)$, which is a nonclassical diffusion equation (diffusion equation with memory). Let $m(t)=m_0{t}^{-\alpha }/\mathrm{\Gamma }(\alpha )$ with $m_0>0$, then one finds that (1.1) is the well-known Rayleigh-Stokes equation:

$${\partial }_{t}u-(1+m_0{\partial }_t^{\alpha })\mathrm{\Delta }u=f(x)g(t)+h(u).$$

This type of equations was constituted in [6], [16] to describe the behavior of some non-Newtonian fluids. Various numerical methods have been developed for forward problem with Rayleigh-Stokes type equations in [1], [2], [4], [3], [15], [20]. We also mention some results on analytic representation of solutions for Rayleigh-Stokes equations in [8], [16], [22], and the recent results on solvability and regularity in [9], [12], [21]. Especially, the terminal value problem related to Rayleigh-Stokes equations has been studied in [10], [19], in which the objective is to recover initial value of the state function from terminal observations.

It should be noted that the problem of recovering source term for Rayleigh-Stokes equations with nonlinear perturbations has not been addressed in literature. In the linear case when $h=0$ and $\phi =\phi (x)$ (independent of u), a result on recovering f has been given in [18]. Our aim in this note is to analyze a circumstance where problem (1.1)-(1.4) is uniquely solvable and the obtained solution is regular and stable with respect to input data. In our setting, the nonlinearities h and φ are allowed to be superlinear, which makes the problem much complicated, from the technical point of view. Our analysis is based on the theory of completely positive functions, local estimates on Hilbert scales, and the fixed point argument.

Mentioning the highlights of our work, one sees the following:

• The problem of identifying source term is carried out in the case that a nonlinear perturbation involves and the final measurement is implicit. This is a practical situation which results in significant difficulties for our analysis.

• Under a reasonable setting, we are able to show that the mild solution of our problem is strong. This feature is useful for establishing numerical schemes in further studies.

Our work is organized as follows. In the next section, we find a representation for solution of (1.1)-(1.4). Moreover, we prove some regularity properties of the resolvent operators. In Section 3, we show the existence and stability results. The last section is devoted to analyzing the regularity of solution, namely, we verify that the mild solution is a strong one.