Bounding the solutions of parametric weakly coupled second-order semilinear parabolic partial differential equations

In this paper, two novel techniques for bounding the solutions of parametric weakly coupled second‐order semilinear parabolic partial differential equations are developed. The first provides a theorem to construct interval bounds, while the second provides a theorem to construct lower bounds convex and upper bounds concave in the parameter. The convex/concave bounds can be significantly tighter than the interval bounds because of the wrapping effect suffered by interval analysis in dynamical systems. Both types of bounds are computationally cheap to construct, requiring solving auxiliary systems twice and four times larger than the original system, respectively. An illustrative numerical example of bound construction and use for deterministic global optimization within a simple serial branch‐and‐bound algorithm, implemented numerically using interval arithmetic and a generalization of McCormick's relaxation technique, is presented. Problems within the important class of reaction‐diffusion systems may be optimized with these tools.