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We investigate a class of parametric elliptic semilinear partial differentialequations of second order with homogeneous essential boundary conditions, wherethe coefficients and the right-hand side (and hence the solution) may depend ona parameter. This model can be seen as a reaction-diffusion problem with apolynomial nonlinearity in the reaction term. The efficiency of variousnumerical approximations across the entire parameter space is closely relatedto the regularity of the solution with respect to the parameter. We show thatif the coefficients and the right-hand side are analytic or Gevrey classregular with respect to the parameter, the same type of parametric regularityis valid for the solution. The key ingredient of the proof is the combinationof the alternative-to-factorial technique from our previous work [1] with anovel argument for the treatment of the power-type nonlinearity in the reactionterm. As an application of this abstract result, we obtain rigorous convergenceestimates for numerical integration of semilinear reaction-diffusion problemswith random coefficients using Gaussian and Quasi-Monte Carlo quadrature. Ourtheoretical findings are confirmed in numerical experiments.

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