Analytic and Gevrey class regularity for parametric semilinear reaction-diffusion problems and applications in uncertainty quantification

TL;DR

This paper proves that solutions to parametric semilinear reaction-diffusion PDEs inherit the regularity of coefficients, enabling efficient numerical integration methods like Quasi-Monte Carlo for uncertainty quantification.

Contribution

It establishes the Gevrey and analytic regularity of solutions with respect to parameters for a class of semilinear PDEs, extending previous techniques to nonlinear reaction terms.

Findings

Solutions are Gevrey or analytic in parameters if coefficients are.

Convergence estimates for numerical integration methods are rigorously derived.

Numerical experiments confirm theoretical regularity results.

Abstract

We investigate a class of parametric elliptic semilinear partial differential equations of second order with homogeneous essential boundary conditions, where the coefficients and the right-hand side (and hence the solution) may depend on a parameter. This model can be seen as a reaction-diffusion problem with a polynomial nonlinearity in the reaction term. The efficiency of various numerical approximations across the entire parameter space is closely related to the regularity of the solution with respect to the parameter. We show that if the coefficients and the right-hand side are analytic or Gevrey class regular with respect to the parameter, the same type of parametric regularity is valid for the solution. The key ingredient of the proof is the combination of the alternative-to-factorial technique from our previous work [1] with a novel argument for the treatment of the power-type nonlinearity in the reaction term. As an application of this abstract result, we obtain rigorous convergence estimates for numerical integration of semilinear reaction-diffusion problems with random coefficients using Gaussian and Quasi-Monte Carlo quadrature. Our theoretical findings are confirmed in numerical experiments.