An inverse source problem for generalized Rayleigh-Stokes equations involving superlinear perturbations
Introduction
Let be a bounded domain with smooth boundary ∂Ω. Consider the following problem: find satisfying where takes values in , g, h and φ are given functions, , stands for the nonlocal differential operator of Riemann-Liouville type defined by with respect to the kernel function .
The aforementioned system is an inverse problem in the situation that the source term 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, can be interpreted as the spatial distribution of source/force which is supposed to be unknown, meanwhile 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 ), (1.1) is classical diffusion equation. If m is a regular function, e.g. then our equation reads with and , which is a nonclassical diffusion equation (diffusion equation with memory). Let with , then one finds that (1.1) is the well-known Rayleigh-Stokes equation: 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 and (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.
Section snippets
Preliminaries
In this work, we make use of the standing hypothesis:
- (M)
The function is nonnegative such that the function is completely positive.
- (PC)
There
Solvability and stability
In this section, we first assume that
- (H1)
The function satisfies and is locally Lipschitzian, that is where is a nonnegative function obeying that
- (H2)
is a continuous function such that .
- (H3)
The function satisfies and the locally Lipschitz condition here is a nonnegative function such that
Regularity analysis
In this section, we will show that the mild solution obtained in Section 2 is a strong solution if φ takes more regular values. The definition of strong solution for (1.1)-(1.4) is as follows. Definition 4.1 A pair is called a strong solution to problem (1.1)-(1.4) iff (1.1), (1.3) and (1.4) hold as equations in .
- (M*)
The kernel function m obeys (M) and is a nonincreasing function.
Conclusion
The problem of identifying source in the Rayleigh-Stokes type equation involving nonlinear perturbations is addressed. In our model, the nonlinearity is allowed to be superlinear and the final measurement is supposed to be implicit. The obtained results include the unique solvability, stability and regularity of solution. Especially, if the final datum is regular enough, the solution is strong in the sense that our equation holds in . Our analysis can be used in other nonlocal models such
Acknowledgements
The authors are grateful to the anonymous reviewer for the useful comments and suggestions which lead to an improvement of the representation of the work.
This research is funded by Vietnam's National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2020.07.
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