Global smooth solutions to the spatially periodic Cauchy problem for dissipative nonlinear evolution equations
The authors study the problem \[ \psi_t= -(\sigma- \alpha) \psi- \sigma \theta_x+ \alpha\psi_{xx}, \] \[ \theta_t= -(1-\beta) \theta+ \nu\psi_x+ (\psi\theta)_x +\beta \theta_{xx}, \quad 0\leq x\leq 1 \] with the conditions \[ \begin{aligned} (\psi,\theta) (0,t) & =(\psi, \theta) (1,t),\;(\psi_x, \theta_x) (0,t)= (\psi_x, \theta_x) (1,t),\;0\leq t\leq T,\\ (\psi,\theta) (x,0) & =\bigl( \psi_0 (x), \theta_0(x) \bigr),\;0\leq x\leq 1,\end{aligned} \] \((\alpha,\beta, \sigma\) and \(\nu\) are constants with \(\alpha,\beta\) being positive). Using the energy method, \(L^1\)-estimates and the Leray-Schauder fixed point theorem, they prove the existence and uniqueness of global classical solutions \(\psi, \theta\in C^{2,\delta} ([0,1] \times [0,T]\), where \(C^{2,\delta}\) are parabolic Hölder spaces.
scientific article; zbMATH DE number 1117606