CEU Electronic Theses and Dissertations, 2014
| Author | Ahsan, Muhammad |
|---|---|
| Title | Existence, Regularity and Perturbation Theory for Some Semilinear Differential Equations in Hilbert Spaces |
| Summary | We consider in a real Hilbert space $H$ the Cauchy problem $(P_{0})\colon u^{\prime}(t)+Au(t)+Bu(t)=f(t), \, 0< t < T; \,u(0)=u_0$, where $A\colon D(A)\subset H \to H$ is a maximal monotone linear operator, $B\colon H \to H$ is a Lipschitz monotone (nonlinear) operator, and $f\colon [0,T] \to H$ is a given function. A typical example of problem $(P_{0})$ is the semilinear heat equation when $-A$ is the Laplace operator $\Delta$ with the homogeneous Dirichlet boundary conditions. We associate with problem $(P_{0})$ the following elliptic-like regularizations: $(P_{1}^{\varepsilon}) \colon -\varepsilon u^{\prime \prime}(t)+u^{\ prime}(t)+Au(t) +Bu(t)=f(t), \, 0< t< T; \, u(0)=u_0, \, u(T)=u_T$, and $(P_{2}^{\varepsilon}) \colon -\varepsilon u^{\prime \prime}(t)+u^{\ prime}(t)+Au(t) +Bu(t)=f(t), \, 0< t < T; \, u(0)=u_0, \, u^{\prime}(T)=u_T$, where $\varepsilon >0$ is a small parameter. Problems $(P_{1}^{\varepsilon})$ and $(P_{2}^{\varepsilon})$ are essentially different and require different methods of investigation. We discuss the existence, uniqueness and higher regularity for the solutions of problems $(P_{0})$, $(P_{1}^{\varepsilon})$ and $(P_{2}^{\varepsilon})$. Then we establish the asymptotic expansions of order zero for the solutions of problems $(P_{1}^{\varepsilon})$ and $(P_{2}^{\varepsilon})$, as well as an asymptotic expansion of order one for the solution of problem $(P_{2}^{\varepsilon})$. A boundary layer of order zero occurs in problem $(P_{1}^{\varepsilon})$ with respect to the norm of $C([0,T];H)$, but the boundary layer of order zero is not visible with respect to the norm of $L^{2}(0,T;H)$. Problem $(P_{2}^{\varepsilon})$ turns out to be a regular perturbation problem of order zero with respect to the norm of $C([0,T];H)$, hence, it is also a regular perturbation problem of order zero with respect to the norm of $L^{2}(0,T;H)$. However, when we establish the asymptotic expansion of order one for the solution of problem $(P_{2}^{\varepsilon})$, a boundary layer of order one occurs with respect to the norm of $C([0,T];H)$, but this boundary layer is not visible with respect to the norm of $L^{2}(0,T;H)$. |
| Supervisor | Morosanu, Gheorghe |
| Department | Mathematics PhD |
| Full text | https://www.etd.ceu.edu/2014/ahsan_muhammad.pdf |
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