Long-time asymptotics for fully nonlinear homogeneous parabolic equations |
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Authors: | Scott N Armstrong Maxim Trokhimtchouk |
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Institution: | 1. Department of Mathematics, University of California, Berkeley, CA, 94720, USA 2. Department of Mathematics, Louisiana State University, Baton Rouge, LA, 70803, USA
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Abstract: | We study the long-time asymptotics of solutions of the uniformly parabolic equation $$ u_t + F(D^2u) = 0 \quad{\rm in}\, {\mathbb{R}^{n}}\times \mathbb{R}_{+},$$ for a positively homogeneous operator F, subject to the initial condition u(x, 0) = g(x), under the assumption that g does not change sign and possesses sufficient decay at infinity. We prove the existence of a unique positive solution Φ+ and negative solution Φ?, which satisfy the self-similarity relations $$\Phi^\pm (x,t) = \lambda^{\alpha^\pm}\Phi^\pm ( \lambda^{1/2} x,\lambda t ).$$ We prove that the rescaled limit of the solution of the Cauchy problem with nonnegative (nonpositive) initial data converges to ${\Phi^+}$ ( ${\Phi^-}$ ) locally uniformly in ${\mathbb{R}^{n} \times \mathbb{R}_{+}}$ . The anomalous exponents α+ and α? are identified as the principal half-eigenvalues of a certain elliptic operator associated to F in ${\mathbb{R}^{n}}$ . |
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