[26.4.2016] Green’s function and infinite-time bubbling in the critical nonlinear heat equation

In this paper of Carmen Cortazar, Manuel del Pino, Monica Musso they studied the profile of solutions which blow-up in in-finite time of the critical nonlinear heat equation

u_t - \Delta u = u^{\frac{n+2}{n-2}}

in a bounded domain \Omega\subset \mathbb R^n with n\geq 5.

The solutions are shown to have bubbling type of behaviour, that means, there exist k points in \Omega which are the only blow-up points of such a solution.

Moreover, such a solution can be constructed approximately by: let q_1, q_2, \ldots, q_k be the bubble points, then

u(x,t) \approx \sum\limits_{j=1}^{k}\alpha_n\left(\dfrac{\mu_j(t)}{\mu_j(t)^2 + |x- \xi_j(t)|^2}\right)^{(n-2)/2}

where \xi_j(t) \rightarrow q_j and 0<\mu_j(t) \rightarrow 0 as t\rightarrow +\infty.

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About baotangquoc

Lecturer School of Applied Mathematics and Informatics Hanoi University of Science and Technology No 1, Dai Co Viet Street, Hanoi
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One Response to [26.4.2016] Green’s function and infinite-time bubbling in the critical nonlinear heat equation

  1. Pingback: [29.4.2016] Dynamics near the ground state for the energy critical nonlinear heat equation in large dimensions | Tăng Quốc Bảo

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