Qualitative theory for a model of laminar flames with arbitrary nonnegative initial data
The system of nonlinear parabolic equations \[ (1)\quad T_ t=d_ 0T_{xx}+QYf(T),\quad Y_ t=d_ 1Y_{xx}-Yf(T) \] models, for certain choices of f, the dynamics of laminar flames in a premixed reactive gas. With \(t\geq 0\), \(-\infty <x<\infty\), \(T=T(x,t)\) denotes the temperature and \(Y=Y(x,t)\) denotes the concentration of the reactant A in the one- step reaction \(A\to B\); the parameters \(d_ 0,d_ 1\) and Q are positive constants. For physical background behind equation (1) see \textit{F. William} [Combustion Theory (Addison-Wesley, Reading, MA 1985)]. The author considers the initial-value problem for (1) \[ (2)\quad T(x,0)=T_ 0(x),\quad Y(x,0)=Y_ 0(x) \] where \(T_ 0(x)\), \(Y_ 0(x)\) are nonnegative and uniformly continuous functions on \({\mathbb{R}}\) and f is a \(C^ 1\)-map from \({\mathbb{R}}\) to the nonnegative reals. N and M are constants such that \(0\leq f(T)\leq M\) for \(T\geq 0\), \(| f'(T)| \leq N\) for all T. Under these conditions he proves the existence of unique global strong solutions T and Y of (1), (2) satisfying \[ \| T(t)\|_{\infty}\leq \| T_ 0\|_{\infty}+Q\| Y_ 0\|_{\infty}Mt;\quad \| Y(t)\|_{\infty}\leq \| Y_ 0\|_{\infty}. \] With additional assumptions on f and \(T_ 0\) he establishes a bound on T and the uniform convergence of Y to the physical situations and practical relevance.
scientific article; zbMATH DE number 4171495