With this article, we wish to honor the many contributions of our mentor, colleague and dear friend, Professor W. Edward Olmstead, on the occasion of his retirement from Northwestern University. Ed has spent over five decades at Northwestern University, first as a graduate student and then as a member of the faculty. During this time he completed his PhD, played a key role in the formation of the Department of Engineering Science and Applied Mathematics (ESAM), developed several courses in applied mathematics, participated in the education of numerous students, and made vast and important contributions in the field of applied mathematics. Here, we give a (mostly chronological) account of some of Ed's major research interests and contributions, primarily in the field of Integral Equations.

In this article, we investigate the local existence of a unique mild solution to a reaction diffusion system with time-nonlocal nonlinearities of exponential growth. Moreover, blowing-up solutions are shown to exist, and their time blow-up profile is presented.

By using Green's function, the problem is converted into an integral equation. It is shown that there exists a $t_b$ such that, for $0\leq t\lt t_b$, the integral equation has a unique nonnegative continuous solution $u$; if $t_b$ is finite, then $u$ is unbounded in $[0, t_b)$. Then, $u$ is proved to be the solution of the original problem.

We consider the Cauchy problem in $\mathbb {R}^N$, $N \geq 1$, for the semi-linear Schr\"odinger equation with fractional Laplacian. We present the local well-posedness of solutions in $H^{{\alpha }/{2}}(\mathbb {R}^N)$, $0\lt \alpha \lt 2$. We prove a finite-time blow-up result, under suitable conditions on the initial data.

In this paper, we consider a Cauchy problem for a nonlinear viscoelastic equation with delay. Under suitable conditions on the initial data and the relaxation function, in the whole space, we prove a finite-time blow-up result.

In previous work, Lo and Tatar studied the exponential decay for a laminated beam with viscoelastic damping acting on the effective rotation angle in the case of equal-speed wave propagations. In this paper, we continue consideration of the same problem in the case of non-equal wave speeds. In this case, the main difficulty is how to estimate the non-equal speed term. To overcome this difficulty, the second-order energy method introduced in Guesmia and Messaoudi seems to be the best choice for our problem. For a wide class of relaxation functions, we establish the general decay result for the energy without any kind of internal or boundary control.

This paper is concerned with the following memory-type Timoshenko system \[ \rho _1\varphi _{tt}-K(\varphi _x+\psi )_x=0 \] \[ \rho _2\psi _{tt}-b\psi _{xx}+K(\varphi _x+\psi )+ \displaystyle \int _0^tg(t-s)\psi _{xx}(s)\,ds=0, \] $(x,t)\in (0,L)\times (0,\infty )$, with Dirichlet boundary conditions, where $g$ is a positive non-increasing function satisfying, for some constant $1\leq p\lt {3}/{2}$, \[ g'(t)\leq -\xi (t)g^p(t),\quad \mbox {for all }t\geq 0. \] We prove some decay results which generalize and improve many earlier results in the literature. In particular, our result gives the optimal decay for the case of polynomial stability.

In this paper, a system of nonlinear integral equations related to combustion problems is considered. Necessary and sufficient conditions for the existence and explosion of positive solutions are given. In addition, the uniqueness of the positive solutions is shown. The main results are obtained by monotonicity methods.

In this paper, we investigate a system of nonlinear integral equations that has previously been proposed in modelling of the human cornea. The main result of our work is a construction of lower and upper estimates that bound the components of the exact solution to the system being considered. These results generalize some of the recent work by other authors. We conclude the paper with a numerical verification of our analytical estimates.

Based on the study of blow up of a particular system of ordinary differential equations, we give a sufficient condition for blow up of positive mild solutions to the Cauchy problem of a fractional reaction-diffusion system, and, by a comparison between the transition densities of the semigroups generated by $\Delta _\alpha :=-(-\Delta )^{\alpha /2}$ and $\Delta _\alpha +b(x)\cdot \nabla $ for $1\lt \alpha \lt 2$, $d\geq 1$ and $b$ in the Kato class on $\mathbb {R}^d$, we prove that this condition is also sufficient for the blow up of a fractional diffusion-convection-reaction system.

In this paper, a split-step collocation method is proposed for solving linear stochastic Volterra integral equations (SVIEs) with smooth kernels. The H\"older condition and the conditional expectations of the exact solutions are investigated. The solvability and mean-square boundedness of numerical solutions are proved and the strong convergence orders of collocation solutions and iterated collocation solutions are also shown. In addition, numerical experiments are provided to verify the conclusions.