| Summary: | In this paper, we analyze the well-posedness of the Cauchy–Dirichlet problem to an integro-differential equation on a multidimensional domain <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Ω</mo><mo>⊂</mo><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></mrow></semantics></math></inline-formula> in the unknown <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>=</mo><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="bold">D</mi><mrow><mi>t</mi></mrow><msub><mi>ν</mi><mn>0</mn></msub></msubsup><mrow><mo>(</mo><msub><mi>ϱ</mi><mn>0</mn></msub><mi>u</mi><mo>)</mo></mrow><mo>−</mo><msubsup><mi mathvariant="bold">D</mi><mrow><mi>t</mi></mrow><msub><mi>ν</mi><mn>1</mn></msub></msubsup><mrow><mo>(</mo><msub><mi>ϱ</mi><mn>1</mn></msub><mi>u</mi><mo>)</mo></mrow><mo>−</mo><msub><mi mathvariant="script">L</mi><mn>1</mn></msub><mi>u</mi><mo>−</mo><msubsup><mo>∫</mo><mrow><mn>0</mn></mrow><mi>t</mi></msubsup><mi mathvariant="script">K</mi><mrow><mo>(</mo><mi>t</mi><mo>−</mo><mi>s</mi><mo>)</mo></mrow><msub><mi mathvariant="script">L</mi><mn>2</mn></msub><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>s</mi><mo>)</mo></mrow><mi>d</mi><mi>s</mi><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>+</mo><mi>g</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>,</mo><mspace width="0.166667em"></mspace><mn>0</mn><mo><</mo><msub><mi>ν</mi><mn>1</mn></msub><mo><</mo><msub><mi>ν</mi><mn>0</mn></msub><mo><</mo><mn>1</mn><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="bold">D</mi><mrow><mi>t</mi></mrow><msub><mi>ν</mi><mi>i</mi></msub></msubsup></semantics></math></inline-formula> are the Caputo fractional derivatives, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ϱ</mi><mi>i</mi></msub><mo>=</mo><msub><mi>ϱ</mi><mi>i</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ϱ</mi><mn>0</mn></msub><mo>≥</mo><msub><mi>μ</mi><mn>0</mn></msub><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">L</mi><mi>i</mi></msub></semantics></math></inline-formula> are uniform elliptic operators with time-dependent smooth coefficients. The principal feature of this equation is related to the integro-differential operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="bold">D</mi><mrow><mi>t</mi></mrow><msub><mi>ν</mi><mn>0</mn></msub></msubsup><mrow><mo>(</mo><msub><mi>ϱ</mi><mn>0</mn></msub><mi>u</mi><mo>)</mo></mrow><mo>−</mo><msubsup><mi mathvariant="bold">D</mi><mrow><mi>t</mi></mrow><msub><mi>ν</mi><mn>1</mn></msub></msubsup><mrow><mo>(</mo><msub><mi>ϱ</mi><mn>1</mn></msub><mi>u</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, which (under certain assumption on the coefficients) can be rewritten in the form of a generalized fractional derivative with a non-positive kernel. A particular case of this equation describes oxygen delivery through capillaries to tissue. First, under proper requirements on the given data in the linear model and certain relations between <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ν</mi><mn>0</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ν</mi><mn>1</mn></msub></semantics></math></inline-formula>, we derive a priori estimates of a solution in Sobolev–Slobodeckii spaces that gives rise to providing the Hölder regularity of the solution. Exploiting these estimates and constructing appropriate approximate solutions, we prove the global strong solvability to the corresponding linear initial-boundary value problem. Finally, obtaining a priori estimates in the fractional Hölder classes and assuming additional conditions on the coefficients <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ϱ</mi><mn>0</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ϱ</mi><mn>1</mn></msub></semantics></math></inline-formula> and the nonlinearity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>g</mi><mo>(</mo><mi>u</mi><mo>)</mo></mrow></semantics></math></inline-formula>, the global one-valued classical solvability to the nonlinear model is claimed with the continuation argument method.
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