By Fernando Casas, Vicente Martínez (eds.)

The publication encompasses a number of contributions given on the 23th Congress on Differential Equations and purposes (CEDYA) / thirteenth Congress of utilized arithmetic (CMA) that happened at Castellon, Spain, in 2013. CEDYA is well known because the congress of the Spanish Society of utilized arithmetic (SEMA) and constitutes the most discussion board and assembly aspect for utilized mathematicians in Spain. The papers integrated during this publication were chosen after a radical refereeing approach and supply a great precis of the new task built through diversified teams operating usually in Spain on purposes of arithmetic to numerous fields of technology and expertise. the aim is to supply an invaluable reference of educational and commercial researchers operating within the quarter of numerical research and its applications.

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T /-weak– and almost everywhere in ˝T . 3˛/. ˝//. Finally, by making k ! 1 in (14) and (15) we obtain (8) and (9). T. González Montesinos and F. ˝T /. ˝T /-weak– . We can 2 pass to the limit in (19) to obtain (12). Remark 4 Our future work consists in establishing under what conditions on can we assure that D 0 or, in other words, how can one derive the strong convergence 1=2 r'k ! ˝T /. Âk / Remark 5 The analysis of the uniqueness of a solution to (6)–(12) is a very complex task even if we already know that D 0.

Casas, V. 1007/978-3-319-06953-1__6 53 54 A. Rodríguez-Bernal and S. Sastre-Gómez 1 Introduction Let ˝ RN be an open, bounded set. x; y/ is a nonnegative function defined as J W ˝ ˝ ! R. ˝//: The function f is defined as f W ˝ R ! x; s/. The equations like (2) have been mainly used to model diffusion processes, as we can see in [1, 3–5]. x; t/ is the rate at which individuals are arriving to position x from all other places, and f is the rate of local reaction. We will start giving a result of existence and uniqueness of the solution associated to (2), with f globally Lipschitz.

RN / with ˛; ˇ 2 R; ˛ ˇ. R / for any 0 Ä 0t Mq;r e t N 4 . RN / q > 0 and 1 < q Ä r Ä 1 and some Mq;r > 0. R // to be specified below, where s; 0 and s C < m where m is the order of the main operator. t /P . / d ; t > 0; (12) 0 with u0 to be chosen below, and where S. / can be S . / or S 2 . /. RN / with s s 2 . RN / Ä Ct 0/ . RN / U where 0 and 0 2 Œ m ; 1 m /. See [8] for details. x/D a u with d 2 L U denotes any derivative of order a, which satisfies the above assumptions for some s and . RN /, a 2 f0; 1; 2; 3g.