Locally uniform convergence to an equilibrium for nonlinear parabolic equations on \(\mathbb{R}^N\).

*(English)*Zbl 1336.35207This paper addresses the asymptotic behavior as \(t\rightarrow \infty\) of bounded solutions to the Cauchy problem

\[ u_t-\Delta u=f(u),\,\, x\in\mathbb{R}^n,\,t>0, \quad u(0,x)=u_0(x),\quad x\in\mathbb{R}^n, \]

where \(u_0\) is a non-negative function with compact support and \(f\) is a function in \(C^1(\mathbb{R})\) satisfying \(f(0)=0\). It is also assumed that \(f'\) is locally Hölder continuous and that \(f\) satisfies a nondegeneracy condition at a specific subset of its zero points.

It is known that nonconstant equilibria for such equations occur in continua, leading to a complicated set of equilibria with a huge variety of elements. Despite this fact, the main results in this paper show that the compactness assumption on \(u_0\) is powerful enough to guarantee that the \(\omega\)-limit set of any bounded solution has a rather simple structure. It consists of a unique equilibrium that is either constant or radially symmetric and decreasing around a fixed point, converging to a constant equilibrium as \(|x|\rightarrow \infty\).

\[ u_t-\Delta u=f(u),\,\, x\in\mathbb{R}^n,\,t>0, \quad u(0,x)=u_0(x),\quad x\in\mathbb{R}^n, \]

where \(u_0\) is a non-negative function with compact support and \(f\) is a function in \(C^1(\mathbb{R})\) satisfying \(f(0)=0\). It is also assumed that \(f'\) is locally Hölder continuous and that \(f\) satisfies a nondegeneracy condition at a specific subset of its zero points.

It is known that nonconstant equilibria for such equations occur in continua, leading to a complicated set of equilibria with a huge variety of elements. Despite this fact, the main results in this paper show that the compactness assumption on \(u_0\) is powerful enough to guarantee that the \(\omega\)-limit set of any bounded solution has a rather simple structure. It consists of a unique equilibrium that is either constant or radially symmetric and decreasing around a fixed point, converging to a constant equilibrium as \(|x|\rightarrow \infty\).

##### MSC:

35K91 | Semilinear parabolic equations with Laplacian, bi-Laplacian or poly-Laplacian |

35K15 | Initial value problems for second-order parabolic equations |

35K05 | Heat equation |

35B40 | Asymptotic behavior of solutions to PDEs |