Asymptotic properties of a reaction-diffusion equation with degenerate \(p\)-Laplacian
The authors study the following reaction-diffusion equation \[ u_t = \lambda(|u_x|^{p-2}u_x)_x + |u|^{q-2}u(1 -|u|^r), \qquad (x,t)\in (0,1)\times (0,+\infty), \] \[ u(0,t) = u(1,t) = 0,\qquad t\in (0,+\infty), \] with initial condition \[ u(x,0) = u_0(x),\qquad x\in (0,1), \] where \(p > 2\), \(q\geq 2\), \(r > 0\) and \(\lambda\) is a positive parameter. They discuss the following: the global existence of solutions; the structure of the set of all stationary solutions; the stability of each stationary solution. The existence and uniqueness of global solutions \(u(\cdot;u_0)\) with \(u(0) = u_0\in L^2\) is proven by applying an abstract result for nonlinear evolution equations. The authors show that the family of solution operators \(\{S(t):u_0\mapsto u(t;u_0)\); \(t\geq 0\}\) has the semigroup property; so that the problem under consideration generates a dynamical system on \(L^2\). In order to investigate the asymptotic behavior of solutions, it is considered the \(\omega\)-limit set \(\omega(u_0)\) associated with \(\{S(t)u_0; t\geq 0\}\). The authors prove that \(\omega(u_0)\subset E_{\lambda}\), where \(E_{\lambda}\) is the set of stationary solutions. The study for \(E_{\lambda}\) is done by the time-map method. The relation between \(p\) and \(q\) makes a remarkable difference on the shape of the time-map and, in consequence, the structure of \(E_{\lambda}\). A rough content of the result is as follows: if \(p > q\), then \(E_{\lambda}\) is an infinite set for any \(\lambda > 0\); if \(p = q\), then \(E_{\lambda}\) is a finite set for large \(\lambda\) and an infinite set for small \(\lambda\) with bifurcations from the trivial solution; if \(p < q\), then \(E_{\lambda}\) is a finite set for large \(\lambda\) and an infinite set for small \(\lambda\) with spontaneous bifurcations. Furthermore, there always exist solutions with flat core for sufficiently small \(\lambda\).
structure of the set of all stationary solutions
scientific article; zbMATH DE number 1491637