TY - JOUR

T1 - Interaction between nonlinear diffusion and geometry of domain

AU - Magnanini, Rolando

AU - Sakaguchi, Shigeru

N1 - Funding Information:
✩ This research was partially supported by a Grant-in-Aid for Scientific Research (B) (♯ 20340031) of Japan Society for the Promotion of Science and by a Grant of the Italian MURST. * Corresponding author. E-mail addresses: magnanin@math.unifi.it (R. Magnanini), sakaguch@amath.hiroshima-u.ac.jp (S. Sakaguchi).

PY - 2012/1/1

Y1 - 2012/1/1

N2 - Let Ω be a domain in RN, where N≥2 and ∂Ω is not necessarily bounded. We consider nonlinear diffusion equations of the form ∂tu=δφ(u). Let u=u(x,t) be the solution of either the initial-boundary value problem over Ω, where the initial value equals zero and the boundary value equals 1, or the Cauchy problem where the initial data is the characteristic function of the set RN\Ω. We consider an open ball B in Ω whose closure intersects ∂ Ω only at one point, and we derive asymptotic estimates for the content of substance in B for short times in terms of geometry of Ω. Also, we obtain a characterization of the hyperplane involving a stationary level surface of u by using the sliding method due to Berestycki, Caffarelli, and Nirenberg. These results tell us about interactions between nonlinear diffusion and geometry of domain.

AB - Let Ω be a domain in RN, where N≥2 and ∂Ω is not necessarily bounded. We consider nonlinear diffusion equations of the form ∂tu=δφ(u). Let u=u(x,t) be the solution of either the initial-boundary value problem over Ω, where the initial value equals zero and the boundary value equals 1, or the Cauchy problem where the initial data is the characteristic function of the set RN\Ω. We consider an open ball B in Ω whose closure intersects ∂ Ω only at one point, and we derive asymptotic estimates for the content of substance in B for short times in terms of geometry of Ω. Also, we obtain a characterization of the hyperplane involving a stationary level surface of u by using the sliding method due to Berestycki, Caffarelli, and Nirenberg. These results tell us about interactions between nonlinear diffusion and geometry of domain.

KW - Cauchy problem

KW - Geometry of domain

KW - Initial behavior

KW - Initial-boundary value problem

KW - Nonlinear diffusion

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U2 - 10.1016/j.jde.2011.08.017

DO - 10.1016/j.jde.2011.08.017

M3 - Article

AN - SCOPUS:80054100364

SN - 0022-0396

VL - 252

SP - 236

EP - 257

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 1

ER -