Long-time asymptotics for fully nonlinear homogeneous parabolic equations
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Long-time asymptotics for fully nonlinear homogeneous parabolic equations

  • Author(s): Armstrong, Scott N.
  • Trokhimtchouk, Maxim
  • et al.
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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