We show that, under so called controllable growth conditions, any weak solution in the energy class of the semilinear parabolic system $u_t (t, x) + Au(t, x) = f (t,x,u, . . . ,∇^m u), (t, x)\in (0, T ) \times \Omega$, is locally regular. Here, A is an elliptic matrix differential operator of order 2m. The result is proved by writing the system as a system with linear growth in $u, . . . ,∇^m u$ but with “bad” coefficients and by means of a continuity method, where the time serves as parameter of continuity. We also give a partial generalization of previous work of the second author and von Wahl to Navier boundary conditions.

Local regularity of weak solutions of semilinear parabolic systems with critical growth / Berchio, Elvise; H. C., Grunau. - In: JOURNAL OF EVOLUTION EQUATIONS. - ISSN 1424-3199. - STAMPA. - 7:(2007), pp. 177-196. [10.1007/s00028-007-9998-2]

### Local regularity of weak solutions of semilinear parabolic systems with critical growth

#### Abstract

We show that, under so called controllable growth conditions, any weak solution in the energy class of the semilinear parabolic system $u_t (t, x) + Au(t, x) = f (t,x,u, . . . ,∇^m u), (t, x)\in (0, T ) \times \Omega$, is locally regular. Here, A is an elliptic matrix differential operator of order 2m. The result is proved by writing the system as a system with linear growth in $u, . . . ,∇^m u$ but with “bad” coefficients and by means of a continuity method, where the time serves as parameter of continuity. We also give a partial generalization of previous work of the second author and von Wahl to Navier boundary conditions.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11583/2522488
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