Submitted by MARCO ANTONIO MOYA ROSAS null (23689400813) on 2016-06-08T23:48:22Z No. of bitstreams: 1 Marco.pdf: 3930307 bytes, checksum: fa27a7c6ffca34b6c67e9ba1944a88be (MD5) / Approved for entry into archive by Juliano Benedito Ferreira (julianoferreira@reitoria.unesp.br) on 2016-06-13T14:55:05Z (GMT) No. of bitstreams: 1 rosas_mm_me_rcla.pdf: 3930307 bytes, checksum: fa27a7c6ffca34b6c67e9ba1944a88be (MD5) / Made available in DSpace on 2016-06-13T14:55:05Z (GMT). No. of bitstreams: 1 rosas_mm_me_rcla.pdf: 3930307 bytes, checksum: fa27a7c6ffca34b6c67e9ba1944a88be (MD5) Previous issue date: 2016-04-28 / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / Neste trabalho, analisaremos o problema de boa colocação do problema de valor inicial para a equação semilinear do calor. Mostraremos a existência de solução global mild, quando o dado inicial u_0 pertence ao espaço L^( n(ρ−1) /2) −fraco e tem norma suficientemente pequena. / In this work, we discuss the well−posedness of the initial value problem for the semilinear heat equation. We show the existence of global mild solution, when the initial data u_0 belong to weak L^(n(ρ−1)/ 2) space with a sufficiently small norm.

Dans cette thèse, nous étudions un problème de contrôle et un problème inverse pour les équationsde la chaleur. Notre premier travail concerne la contrôlabilité à zéro pour une équation de la chaleur semi-linéaire. Il est à noter que sans contrôle, la solution est instable et il y aura en général explosion de la solution en un temps fini. Ici, nous proposons un résultat positif de contrôlabilité à zéro sous une hypothèse quantifiée de petitesse sur la donnée initiale. La nouveauté réside en la construction de ce contrôle pour amener la solution à l’état d’équilibre.Notre second travail aborde l’équation de la chaleur rétrograde dans un domaine borné et sous la condition de Dirichlet. Nous nous intéressons à la question suivante: peut-on reconstruire la donnée initiale à partir d’une observation de la solution restreinte à un sous-domaine et à un temps donné? Ce problème est connu pour être mal-posé. Ici, les deux principales méthodes proposées sont: une approche de filtrage des hautes fréquences et une minimisation à la Tikhonov. A chaque fois, nous reconstruisons de manière approchée la solution et quantifions l’erreur d’approximation / My thesis focuses on two main problems in studying the heat equation: Control problem and Inverseproblem.Our first concern is the null controllability of a semilinear heat equation which, if not controlled, can blow up infinite time. Roughly speaking, it consists in analyzing whether the solution of a semilinear heat equation, underthe Dirichlet boundary condition, can be driven to zero by means of a control applied on a subdomain in whichthe equation evolves. Under an assumption on the smallness of the initial data, such control function is builtup. The novelty of our method is computing the control function in a constructive way. Furthermore, anotherachievement of our method is providing a quantitative estimate for the smallness of the size of the initial datawith respect to the control time that ensures the null controllability property.Our second issue is the local backward problem for a linear heat equation. We study here the followingquestion: Can we recover the source of a linear heat equation, under the Dirichlet boundary condition, from theobservation on a subdomain at some time later? This inverse problem is well-known to be an ill-posed problem,i.e their solution (if exists) is unstable with respect to data perturbations. Here, we tackle this problem bytwo different regularization methods: The filtering method and The Tikhonov method. In both methods, thereconstruction formula of the approximate solution is explicitly given. Moreover, we also provide the errorestimate between the exact solution and the regularized one.

Submitted by HERACLIO LEDGAR LÓPEZ LÁZARO null (herack_11@hotmail.com) on 2016-03-21T12:48:28Z No. of bitstreams: 1 Heracliodissertação.pdf: 1074830 bytes, checksum: eacc291c2e8f474bef30477ea2c47a2f (MD5) / Approved for entry into archive by Juliano Benedito Ferreira (julianoferreira@reitoria.unesp.br) on 2016-03-22T14:20:35Z (GMT) No. of bitstreams: 1 lazaro_hll_me_sjrp.pdf: 1074830 bytes, checksum: eacc291c2e8f474bef30477ea2c47a2f (MD5) / Made available in DSpace on 2016-03-22T14:20:35Z (GMT). No. of bitstreams: 1 lazaro_hll_me_sjrp.pdf: 1074830 bytes, checksum: eacc291c2e8f474bef30477ea2c47a2f (MD5) Previous issue date: 2016-03-07 / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / The problem that we are going to study in this work, is motivated by the dynamics of differential equations nonautonomous. We will establish the existence and uniqueness of solution for a class of parabolic semilineares equations with Dirichlet boundary condition, in a family of domains that varies with time. In addition, certain hypotheses about the non-linearity, we will show the existence of a family of attractors pullback. / O problema que vamos estudar neste trabalho é motivado pela dinâmica de equações diferenciais não autônomas. Vamos estabelecer a existência e unicidade de solução para uma classe de equaçõoes parabólicas semilineares com condição de fronteira de Dirichlet, em uma família de domínios que varia com o tempo. Além disso, sob certas hipóteses sobre a não linearidade, mostraremos a existência de uma família de atratores pullback.