Smoothing effect of absorption for degenerate parabolic equations in 1-d
The author studies the Cauchy and the Dirichlet problems in the domains \((t,x)\in {{\Pi }_{T}}=(0,T)\times \mathbb{R}\) and \((t,x)\in {{Q}_{T}}=(0,T)\times (-l,l)\) respectively, \(T>0\) arbitrary, for the equation \[ {{u}_{t}}={{(k(t,u){{u}_{x}})}_{x}}-f(t,u) \] where \(f(t,0)=k(t,0)=0\), \(k(t,u)\geq 0\), \(f(t,u)\geq 0\) for \(u\geq 0\) and for some \(\alpha \in (0,1)\) we have that \(k(t,u)\in C_{t,u}^{1,1+\alpha }((0,T),(0,\infty ))\) and \(f(t,u)\in {{C}^{\alpha }}((0,T),(0,\infty ))\). The Cauchy problem is considered under the initial condition \(u(0,x)={{u}_{0}}(x),\) for \(x\in \mathbb{R}\), where \({{u}_{0}}(x)\in \mathrm{Lip}(\mathbb{R})\) and \(0\leq {{u}_{0}}(x)\leq M,\;x\in \mathbb{R}\). Then under one additional growth condition the author prove that for any \(T>0\) the Cauchy problem has a global strong solution \(u(t,x)\), with \(0\leq u(t,x)\leq M,\;(t,x)\in {{\Pi }_{T}}\) and \(||{{u}_{x}}|{{|}_{{{L}^{\infty }}({{\Pi }_{T}})}}\leq K\). The Dirichlet problem is considered under the conditions \(u(0,x)={{u}_{0}}(x),\;|x|<l\), \({{u}_{0}}(\pm l)=u(t,\pm l)=0,\text{ }t\in [0,T]\), where \(0\leq {{u}_{0}}(x)\leq M,\;x\in [-l,l]\) and \({{u}_{0}}(x)\in \mathrm{Lip}([-l,l])\). Then the considered Dirichlet problem under some natural growth condition has in \((t,x)\in {{Q}_{T}}=(0,T)\times (-l,l)\) a global strong solution \(u(t,x)\) for any \(T>0\) with \(0\leq u(t,x)\leq M,\;(t,x)\in {{Q}_{T}}\), \(||{{u}_{x}}|{{|}_{{{L}^{\infty }}({{Q}_{T}})}}\leq K\). In additional for the case when no absorption term is present in the equation are proved results of local existence of a strong solution.
scientific article; zbMATH DE number 7001383
scientific article