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Dissertations |
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1
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JOSIAS VERA BACA
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Weighted semilinear Schrödinger equation
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Advisor : CLAUDIANOR OLIVEIRA ALVES
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COMMITTEE MEMBERS :
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BRUNO SERGIO VASCONCELOS DE ARAUJO
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CLAUDIANOR OLIVEIRA ALVES
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FLANK DAVID MORAIS BEZERRA
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Data: Nov 17, 2025
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Show Abstract
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In this work, we use semigroup theory and the energy method, together with compactness
arguments, to establish the local existence of a solution and the well-posedness
of the weighted semilinear Schrödinger equation.
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2
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LARYSSA KELY ALVES RODRIGUES
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Existence of solutions with prescribed norm for a class of semilinear elliptic equations
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Advisor : ROMILDO NASCIMENTO DE LIMA
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COMMITTEE MEMBERS :
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CÉSAR ENRIQUE TORRES LEDESMA
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MARCO AURELIO SOARES SOUTO
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ROMILDO NASCIMENTO DE LIMA
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Data: Nov 21, 2025
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Show Abstract
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In this work, we intend to study the existence of normalized solutions for a class of semilinear elliptic equations of the type $-\Delta u=\lambda u +g(u)$ in $\mathbb{R}^N$, with certain conditions on the function $g$. In recent years, this type of problem has attracted great interest because the norm is preserved from evolution, being relevant for physics. We know that, in $\mathbb{R}^N$, we lose compactness in Sobolev embeddings. Thus, to circumvent this situation, Jeanjean (1997) worked in the space $H^1_{rad}(\mathbb{R}^N)$. Through a variational approach based on Ekeland's Variational Principle and the Mountain Pass, the author establishes the existence of solutions. In this sense, we will present in detail the results obtained by Jeanjean (1997), which is one of the first works to address this topic and widely referenced.
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3
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MARISA DA CUNHA BEZERRA
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Exact Boundary Controllability for the Wave Equation in a Non-Cylindrical Domain
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Advisor : PAMMELLA QUEIROZ DE SOUZA
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COMMITTEE MEMBERS :
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FÁGNER DIAS ARARUNA
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ALDO TRAJANO LOUREDO
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BRUNO SERGIO VASCONCELOS DE ARAUJO
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PAMMELLA QUEIROZ DE SOUZA
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SEVERINO HORACIO DA SILVA
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Data: Nov 24, 2025
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Show Abstract
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In this work, we investigate the problem of exact controllability for the equation
$$u''-\Delta u=0 \quad \text{in} \quad \widehat{Q},$$
where $\hat{Q}$ is a non-cylindrical domain of $\mathbb{R}^{n+1}$. To overcome the challenges imposed by the non-cylindrical geometry, we perform an appropriate change of variables, which allows us to transform problem \eqref{Eq.Onda_} into an equivalent problem in a cylindrical domain. Thus, we study the exact controllability of the equation
$$w'' - \displaystyle\sum_{i,j=1}^n \frac{\partial}{\partial x_i} \left( a_{ij}(x,t) \frac{\partial w}{\partial x_j} \right) + \sum_{i=1}^n b_i(x,t) \frac{\partial w'}{\partial x_i} + \sum_{i=1}^n \beta_i(x,t) \frac{\partial w}{\partial x_i} = 0 \quad \text{in} \quad Q,$$
where $Q=\Omega \times (0,T)$ is a cylinder, with $\Omega \subset \mathbb{R}^n$ a bounded domain. The well-posedness results are established via the Galerkin method, ensuring existence and uniqueness of solutions. In this setting, we study the exact controllability using the Hilbert Uniqueness Method (HUM) and, finally, demonstrate the equivalence between these systems, which ensures that the controllability results obtained in the cylinder extend rigorously to the non-cylindrical domain $\hat{Q}.$
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