Problem of Determining the Reaction Coefficient in a Fractional Diffusion Equation

For a fractional diffusion equation with reaction coefficient depending only on the first two components of the spatial variable $$x=(x_1,x_2,x_3)\in \mathbb {R}^3 $$ and on time $$t\geq 0 $$ , we consider the inverse problem of determining this coefficient under the assumption that the initial value at $$t=0 $$ is known for the solution of the equation and the boundary value at $$ x_3=0$$ is given as an additional condition. This inverse problem is reduced to equivalent integral equations, and we apply the contraction mapping principle to prove the existence of solutions of these equations. Local existence and global uniqueness theorems are proved. We also obtain a stability estimate for the solution of the inverse problem.

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