The non-existence and existence of non-constant positive solutions for a diffusive autocatalysis model with saturation. (English) Zbl 07921261
Summary: This paper deals with a diffusive autocatalysis model with saturation under Neumann boundary conditions. Firstly, some stability and Turing instability results are obtained. Then by the maximum principle, Hölder inequality and Poincaré inequality, a priori estimates and some basic characterizations of non-constant positive solutions are given. Moreover, some non-existence results are presented for three different situations. In particular, we find that the model does not have any non-constant positive solution when the parameter which represents the saturation rate is large enough. In addition, we use the theories of Leray-Schauder degree and bifurcation to get the existence of non-constant positive solutions, respectively. The steady-state bifurcations at both simple and double eigenvalues are intensively studied and we establish some specific condition to determine the bifurcation direction. Finally, a few of numerical simulations are provided to illustrate theoretical results.
MSC:
| 35K57 | Reaction-diffusion equations |
| 35B35 | Stability in context of PDEs |
| 35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |
| 35B45 | A priori estimates in context of PDEs |
Keywords:
autocatalysis; saturation; Turing instability; non-constant positive steady-state solution; bifurcationReferences:
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