| Summary: | This paper presents the method of separation of variables to find conditions on the right-hand side and on the initial data in the Rayleigh-Stokes problem, which ensure the existence and uniqueness of the solution. Further, in the Rayleigh-Stokes problem, instead of the initial condition, the non-local condition is considered: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>T</mi><mo>)</mo><mo>=</mo><mi>β</mi><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo><mo>+</mo><mi>φ</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> is equal to zero or one. It is well known that if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, then the corresponding problem, called the backward problem, is ill-posed in the sense of Hadamard, i.e., a small change in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>T</mi><mo>)</mo></mrow></semantics></math></inline-formula> leads to large changes in the initial data. Nevertheless, we will show that if we consider sufficiently smooth current information, then the solution exists, it is unique and stable. It will also be shown that if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>, then the corresponding non-local problem is well-posed and inequalities of coercive type are satisfied.
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