| Abstract: |
| We will present some mathematical results for a new model coupling the Cahn-Hilllard-Brinkman system with an evolutionary equation describing the active (chemotactic) transport of a chemical species influencing the phase separation process. Specifically, the model may arise in connection with tumor growth processes; mathematically speaking, it may be interesting in itself as it provides a new coupling between a Keller-Segel-like relation (the equation describing the evolution of the concentration of the chemical substance) and a fourth order (rather than a second order as in most models for chemotaxis) evolutionary system. Our main result will be devoted to proving existence of weak solution in the case when the chemotaxis sensitivity function has a controlled growth at infinity; a particular emphasis will be given to discussing the occurrence of critical exponents and to presenting a regularization scheme compatible with the a-priori estimates. |
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