Long-time behaviour of solutions of a class of nonlinear parabolic equations
The long-time behaviour of solutions of the nonlinear parabolic equation of the form: \[ v_t= f(v_x)v_{xx}+ g(v_x),\quad x\in \mathbb R,\;t> 0, \] is studied, under certain conditions on \(f\) and \(g\), which are some \(C^2(\mathbb{R})\) functions. First the result on existence and uniqueness of the following initial-boundary problem: \[ \begin{aligned} & v_t= f(v_x) v_{xx}+ g(v_x),\quad x\in [0,\infty),\;t> 0,\\ & v_x(0,t)= 0,\\ & v_x(x,t)\to m\quad\text{as }x\to\infty,\\ & v(x,0)= v_0(x)\quad(m> 0\text{ is given constant})\end{aligned}\tag{1} \] is stated under some assumptions on \(v_0\). Later a travelling wave solution (it means the solution of the form \(v= w(x)+ ct)\) to the initial-boundary problem (1) is looked for. For the case of \(g''> 0\), the solution of problem (1) is shown to converge to a travelling wave as \(t\to\infty\) in the sense that \[ \sup_{x\geq 0}\,|v(x,t)- (w(x)+ ct)|\to 0\quad\text{as }\to\infty, \] where \(c= g(m)\). For the case of \(g''< 0\), the solution of problem (1) is shown to converge to the similarity solution in the sense that \[ {1\over t}\Biggl| v(x,t)- tP\Biggl({x\over t}\Biggr)\Biggr|\to 0\quad\text{as }t\to\infty, \] uniformly on the sets \(\{(x, t)\in[0,\infty)^2: x\leq Rt\}\) for all \(R> 0\), where similarity solution satisfies the Hamilton-Jacobi equation: \(v_t= g(v_x)\).
scientific article; zbMATH DE number 5382870
scientific article