Existence and uniqueness of a regular solution of the Cauchy-Dirichlet problem for doubly nonlinear parabolic equations
The paper is devoted to the proof of existence and uniqueness of a regular solution of the Cauchy-Dirichlet problem for doubly nonlinear parabolic equations \[ u_t= \text{div } a(u, \nabla u)+ f,\quad (x, t)\in Q_T= \Omega\times (0, T],\;\Omega\in \mathbb{R}^n, \] with initial-boundary conditions \(u= \psi\) on \(\Gamma_T\), where \(a: \mathbb{R}\times \mathbb{R}^n\to \mathbb{R}^n\) is continuous, \(f\in L_1(Q_T)\), \(0\leq \psi\in W^1_1\). It is assumed that: 1) Growth condition: For any \(u\in \overline R_+\) and \(p\in \mathbb{R}^n\) it holds \[ a(u, p) p \geq \nu_0|u|^l |p|^m- \mu_0(|u|^\delta+ 1),\qquad |a(u, p)|\leq \mu_1 |u|^l |p|^{m- 1}+ \mu(|u|) |u|^\alpha, \] where \(\nu_0> 0\), \(\alpha= l/m\), \(\mu(s)\geq 1\) is non-decreasing on \(R_+\), \((m, l)\in D\backslash \omega\), \(D= \{(m, l): m> 1, l\geq 0\}\), \(\omega= \{(m, l)\in D: (\sigma+ 1)/(\sigma+ 2)\leq 1/m- 1/n, \sigma= l/(m- 1)\}\). 2) Strict monotonicity condition: There exists a constant \(\nu_1> 0\) that for any \(u\in \mathbb{R}\) and \(p, q\in \mathbb{R}^n\), \[ (a(u, p)- a(u, q)) (p- q)\geq \nu_1 |u|^l |p- q|^k (|p- b|+ |q- b|)^{1- k/m}, \] where \(b: \mathbb{R}\mapsto \mathbb{R}^n\) is a continuous vector function, \(k= m\) if \(m\geq 2\) and \(k= 2\) if \(1< m< 2\). 3) Local Lipschitz condition: For any \(u, v\in [\varepsilon, M]\) and any \(p\in \mathbb{R}^n\) it holds \(|a(u, p)- a(v, p)|\leq \Lambda |u- v|(1+ |p|^{m- 1})\), where \(\Lambda= \Lambda(\varepsilon, M)\geq 0\). Under the above assumptions existence and uniqueness of a Hölder continuous generalized solution of the Cauchy-Dirichlet problem are proved.
scientific article; zbMATH DE number 856793