On a semilinear parabolic equation arising in optical bistability

Solving the inverse problem of determining an unknown control parameter in a semilinear parabolic equation

The inverse problem of identifying an unknown source control param- eter in a semilinear parabolic equation under an integral overdetermina- tion condition is considered. The series pattern solution of the proposed problem is obtained by using the weighted homotopy analysis method (WHAM). A description of the method for solving the problem and nding the unknown parameter is derived. Finally, tw...

Stable Identification of a Semilinear Term in a Parabolic Equation

We consider a semilinear parabolic equation in a rectangular domain Ω ⊂ R: (∂tu)(x, t) = ∆u(x, t) + a(u(x, t)) with the zero initial value and suitable Dirichlet data. We discuss an inverse problem of determining the nonlinear term a(·) from Neumann data ∂u ∂n on ∂Ω × (0, T ). Under appropriate Dirichlet data, we prove conditional stability of the Hölder type in this inverse problem within a su...

Blow-Up in a Fourth-Order Semilinear Parabolic Equation From . . .

Interior Gradient Blow-up in a Semilinear Parabolic Equation

We present a one dimensional semilinear parabolic equation for which the spatial derivative of solutions becomes unbounded in finite time while the solutions themselves remain bounded. In our example the derivative blows up in the interior of the space interval rather than at the boundary, as in earlier examples. In the case of monotone solutions we show that gradient blow-up occurs at a single...

Optimal control of a semilinear parabolic equation with singular arcs

This paper develops a theory of singular arc, and the corresponding second order necessary and sufficient conditions, for the optimal control of a semilinear parabolic equation with scalar control applied on the r.h.s. We obtain in particular an extension of Kelley’s condition, and the characterization of a quadratic growth property for a weak norm.