On a simple maximum principle technique applied to equations on the circle
The authors consider, on the unit circle \(S^1\), the general quasilinear equation of the form \[ \begin{aligned} \frac{\partial w}{\partial t}(x,t)&= A(w)\cdot w_{xx}+B(w)\cdot w_x+C(w), \\ w( x,0)&= w_0(x), \quad x\in S^1, \end{aligned} \tag{1} \] where \((x,t)\in S^1\times[0,T)\), \(T>0\), \(A(\theta), B(\theta), C(\theta):\mathbb{R} \rightarrow \mathbb{R}\) are given but arbitrary smooth functions, \(w_0(x) \) is a given smooth function on \(S^1\). In order for \((1)\) to be parabolic, the authors consider that for the initial data \( w_0(x)\) the parabolic condition \(A(w_0(x))\geq C>0\) for all \(x\in S^1\), is satisfied, where \(C\) is some positive constant. As a consequence the authors obtain the existence for a unique solution \(w(x,t) \) which is smooth and is defined on \( S^1\times[ 0,T) \) for some finite \(T>0.\) Supposing that \(w\) is bounded on \(S^1\times[0,T)\), the purpose of the paper is to show, using the maximum principle technique, that all higher derivatives (space and time) of \(w\) are also bounded on \( S^1\times[0,T).\) The authors give some specific examples and under some suitable assumptions they derive exponential decay of the derivatives of the solution.
scientific article; zbMATH DE number 5299424