TY - JOUR

T1 - Radial symmetric stationary solutions for a MEMS type reaction–diffusion equation with spatially dependent nonlinearity

AU - Ichida, Yu

AU - Sakamoto, Takashi Okuda

N1 - Funding Information:
The authors would like to express their sincere gratitude to Professor Matsue Kaname (of Kyushu University) for a number of helpful comments. Acknowledgement also go to referees for their careful reading and helpful comments.
Publisher Copyright:
© 2020, The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature.

PY - 2021/2

Y1 - 2021/2

N2 - 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.

AB - 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.

KW - Asymptotic behavior

KW - Desingularization of vector fields (blow-up)

KW - MEMS equation

KW - Poincaré compactification

UR - http://www.scopus.com/inward/record.url?scp=85089995717&partnerID=8YFLogxK

U2 - 10.1007/s13160-020-00438-8

DO - 10.1007/s13160-020-00438-8

M3 - Article

AN - SCOPUS:85089995717

VL - 38

SP - 297

EP - 322

JO - Japan Journal of Industrial and Applied Mathematics

JF - Japan Journal of Industrial and Applied Mathematics

SN - 0916-7005

IS - 1

ER -