Optimal growth rates for a viscous Hamilton-Jacobi equation
The large-time behavior of nonnegative solutions to the Cauchy problem for a viscous Hamilton-Jacobi equation with nonnegative initial data decaying at infinity is investigated. Namely, it is established that the function \(t\to t^{-N/m} | u(t)| _m\) (\(| u(t)| _m\) is the \(L_m\)-norm of the solution \(u(t)\)) is bounded from above and from below by positive real numbers. This result indicates an asymptotic behavior dominated by the hyperbolic part of the considered equation. Some estimates of the change of the value \(\Delta \ln u(t)\) are also established.
scientific article; zbMATH DE number 5010079