Publications of Ahsan, Muhammad

Asymptotic expansions for elliptic-like regularizations of semilinear evolution equations

Consider in a real Hilbert space $H$ the Cauchy problem $(P_{0})\colon u^{\prime}(t)+Au(t)+Bu(t)=f(t), \, 0\leq t \leq T; \, u(0)=u_{0}$, where $-A$ is the infinitesimal generator of a $C_0$-semigroup of contractions, $B$ is a nonlinear monotone operator, and $f$ is a given $H$-valued function. Inspired by the excellent book on singular perturbations by J.L. Lions, we associate with problem $(P_{0})$ the following regularization: $(P_{\varepsilon})\colon -\varepsilon u^{\prime \prime}(t)+u^{\prime}(t)+Au(t)+Bu(t)=f(t), \, 0\leq t \leq T; \,u(0)=u_{0}, \, u^{\prime}(T)=u_{T},$ where $\varepsilon >0$ is a small parameter. We investigate existence, uniqueness and higher regularity for problem $(P_{\varepsilon})$. Then we establish asymptotic expansions of order zero, and of order one, for the solution of $(P_{\varepsilon})$. Problem $(P_{\varepsilon})$ turns out to be regularly perturbed of order zero, and singularly perturbed of order one, with respect to the norm of $C([0,T];H)$. However, the boundary layer of order one is not visible through the norm of $L^{2}(0,T;H)$.

Elliptic-like regularization of semilinear evolution equations

Consider in a real Hilbert space the Cauchy problem (P0): u′(t)+Au(t)+Bu(t) = f (t), 0 ≤ t ≤ T ; u(0) = u_0, where −A is the generator of a C_0-semigroup of linear contractions and B is a smooth nonlinear operator. We associate with (P_0) the following problem: (Pε): −εu′′(t) + u′(t) + Au(t) + Bu(t) = f (t), 0 ≤ t ≤ T ; u(0) = u_0, u(T ) = u_1, where ε > 0 is a small parameter. Existence, uniqueness and higher regularity for both (P0) and (Pε) are investigated and an asymptotic expansion for the solution of problem (Pε) is established, showing the presence of a boundary layer near t = T .