Volume 26, 2020
|Number of page(s)||34|
|Published online||19 February 2020|
Multiplicative controllability for nonlinear degenerate parabolic equations between sign-changing states*
Department of Mathematics and Applications “R. Caccioppoli”, University of Naples Federico II,
** Corresponding author: firstname.lastname@example.org
Accepted: 28 October 2019
In this paper we study the global approximate multiplicative controllability for nonlinear degenerate parabolic Cauchy problems. In particular, we consider a one-dimensional semilinear degenerate reaction-diffusion equation in divergence form governed via the coefficient of the reaction term (bilinear or multiplicative control). The above one-dimensional equation is degenerate since the diffusion coefficient is positive on the interior of the spatial domain and vanishes at the boundary points. Furthermore, two different kinds of degenerate diffusion coefficient are distinguished and studied in this paper: the weakly degenerate case, that is, if the reciprocal of the diffusion coefficient is summable, and the strongly degenerate case, that is, if that reciprocal isn’t summable. In our main result we show that the above systems can be steered from an initial continuous state that admits a finite number of points of sign change to a target state with the same number of changes of sign in the same order. Our method uses a recent technique introduced for uniformly parabolic equations employing the shifting of the points of sign change by making use of a finite sequence of initial-value pure diffusion problems. Our interest in degenerate reaction-diffusion equations is motivated by the study of some energy balance models in climatology (see, e.g., the Budyko-Sellers model) and some models in population genetics (see, e.g., the Fleming-Viot model).
Mathematics Subject Classification: 93C20 / 35K10 / 35K65 / 35K57 / 35K58
Key words: Approximate controllability / bilinear controls / degenerate parabolic equations / semilinear reaction-diffusion equations / sign-changing states
This work was supported by the Istituto Nazionale di Alta Matematica (INdAM), through the GNAMPA Research Project 2016 “Controllo, regolarità e viabilità per alcuni tipi di equazioni diffusive” (coordinator P. Cannarsa), and the GNAMPA Research Project 2017 “Comportamento asintotico e controllo di equazioni di evoluzione non lineari” (coordinator C. Pignotti). Moreover, this research was performed in the framework of the GDRE CONEDP (European Research Group on “Control of Partial Differential Equations”) issued by CNRS, INdAM and Université de Provence. This paper was also supported by the research project of the University of Naples Federico II: “Spectral and Geometrical Inequalities”.
© EDP Sciences, SMAI 2020
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