Radial symmetric stationary solutions for a MEMS type reaction–diffusion equation with spatially dependent nonlinearity

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Abstract

We consider the radial symmetric stationary solutions of ut= Δu- | x| qu-p. We first give a result on the existence of the negative value functions that satisfy the radial symmetric stationary problem on a finite interval for p∈ 2 N, q∈ R. Moreover, we give the asymptotic behavior of u(r) and u(r) at both ends of the finite interval. Second, we obtain the existence of the positive radial symmetric stationary solutions with the singularity at r= 0 for p∈ N and q≥ - 2. In fact, the behavior of solutions for q> - 2 and q= - 2 are different. In each case of q= - 2 and q> - 2 , we derive the asymptotic behavior for r→ 0 and r→ ∞. These facts are studied by applying the Poincaré compactification and basic theory of dynamical systems.

Original languageEnglish
JournalJapan Journal of Industrial and Applied Mathematics
DOIs
Publication statusAccepted/In press - 2020

Keywords

  • Asymptotic behavior
  • Desingularization of vector fields (blow-up)
  • MEMS equation
  • Poincaré compactification

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