Abstract
In this paper, we discuss the influence of assuming L m regularity of initial data, instead of L 1, on a heat or damped wave equation with nonlinear memory. We find that the interplay between the loss of decay rate due to the presence of the nonlinear memory and to the assumption of initial data in L m instead of L 1, leads to a new critical exponent for the problem, whose shape is quite different from the one of the critical exponent for L m theory for the corresponding problem with power nonlinearity |u|p. We prove the optimality of the critical exponent using the test function method.
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Acknowledgements
The author warmly thanks Prof. Michael Reissig who first inspired him to study the influence of a nonlinear memory on evolution equations. Prof. Reissig also proposed to the author many interesting problems related to long time estimates for hyperbolic equations. His friendship and his mathematical intuition have been a great motivation for the author.
The author is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
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D’Abbicco, M. (2021). A New Critical Exponent for the Heat and Damped Wave Equations with Nonlinear Memory and Not Integrable Data. In: Cicognani, M., Del Santo, D., Parmeggiani, A., Reissig, M. (eds) Anomalies in Partial Differential Equations. Springer INdAM Series, vol 43. Springer, Cham. https://doi.org/10.1007/978-3-030-61346-4_9
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DOI: https://doi.org/10.1007/978-3-030-61346-4_9
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