Relative stabilization of a nonlinear degenerate parabolic equation (Q1874827)

The author of this paper considers the Cauchy problem \[ u_t= (u^\alpha)_{xx}+ a(j^\lambda)_x+cv,\;(x,t)\in S=\mathbb{R}^1 \times (0,\infty) \tag{1} \] \[ u(x,0)=f(x),\;x\in\mathbb{R}^1,\tag{2} \] where \(u\in \mathbb{R}^1\) is the state of the perturbed process, \(f\in\mathbb{R}^1\) is the initial perturbed state, \(v\in\mathbb{R}^1\) is the input (control) and \(a,c,\alpha\) and \(\lambda\) are positive numbers with \(\alpha>1\) and \(\lambda<\alpha\). A function \(v(x,t)\), \((x,t)\in\overline S\), belongs to the set \(v\) of admissible controls if it has the following properties: it is continuous in \(\overline S\), satisfies the conditions (a) \(| v(x,t)|\leq u(x,t)\) if \(u(x,t)>1\) and (b) \(| v(x,t)|<1\) if \(u(x,t)\leq 1\), and provides only nonnegative solutions to problem (1), (2). A solution of (1) for a given control \(v(x,t)\) is a nonnegative function \(v(x,t)\) continuous in \(\overline S\) which satisfies the integral identity \[ \int^{t_2}_{t_1} \int_\mathbb{R} \{uh_t+u^\alpha h_{xx}-u^\lambda h_x+cvh\}dx\,dt-\int_\mathbb{R} uh\int^{t_2}_{t_1}dx=0, \] for all \(t_i\) such that \(0<t_1< t_2\) and for an arbitrary nonnegative function \(h(x,t)\in C^{2,1}_{x,t} (S)\) compactly supported in \(S\). Equation (1) is said to be stabilizable with respect to a class \(G\) of initial conditions if, for any function \(f\in G\), there exists an admissible control \(v\in V\) such that the corresponding solution to problem (1), (2) has the property \(\lim_{t\to \infty}u(x,t) =0\) for arbitrary real \(x\). The main result states that if \(\alpha>1\), \((\alpha-\lambda)\in(0,1]\), \((2\lambda-\alpha)\in(0,1)\), then there exists a nonempty class \(G\) (rigorously constructed) with respect to which equation (1) is stabilizable by the control \(v=-ru^\beta\), \(r\in[0,1]\), \(\beta=2\lambda-\alpha\), \(\beta\in(0,1)\). This class \(G\) depends on \(r\), i.e. \(G=G(r)\).