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The linearization principle for some evolution problems arising in chemical kinetics. (Russian, English) Zbl 1043.35104

Sib. Mat. Zh. 43, No. 3, 579-590 (2002); translation in Sib. Math. J. 43, No. 3, 463-472 (2002).
The author studies a mixed problem for a nonlinear system of hyperbolic type arising in the context of mathematical modeling of heat and mass transport in a chemical reactor. In particular, special attention is paid to a mathematical model of chemical processes in a pseudoliquefied layer of a catalyst in which the hyperbolic system is supplemented with an evolution integral equation \[ \begin{gathered} V_t + K(x)V_x = \Phi(x,U),\\ \frac{\partial u_{n+1}}{\partial t} = \int_0^1 f_1(x,V)\,dx + f_2(x,u_{n+1}),\\ I_0V(0,t) + I_1V(1,t) = 0,\\ U(x,0) = U_0(x). \end{gathered} \] Here \(U(x,t) = (V(x,t),u_{n+1}(x,t))\) is the \((n+1)\)-dimensional column vector of unknown functions with \(V = (u_1,\dots,u_n)^T\) and \(U_0(x) = (V_0(x),u_{0_{n+1}}(x))\). The matrices \(K(x)\), \(I_0\), and \(I_1\) and the vector function \(\Phi\) have the former meaning, while the functions \(f_1(x,V)\) and \(f_2(x,u_{n+1})\) are considered as known. The author studies the question of validity of stability theorem in the first approximation for the above evolution system. It is supposed that \(U_c \equiv 0\) is a stationary solution to the problem under study and that the sufficiently smooth functions \(\Phi\), \(f_1\), and \(f_2\) vanish identically on this solution. Linearizing this evolution system at the zero solution, the author obtains a linear problem and studies the behavior of the solution \(U(x,t)\) to this linear problem as \(t\to\infty\).
The main aim is to obtain the following estimate \[ \| U(x,t)\| _{R_t[0,1]} \leq Ke^{-\gamma t}\| U_0(x)\| _{C[0,1]},\quad t > 0, \] for a piecewise smooth solution \(U(x,t)\) of the linearized problem. Derivation of this estimate is the main point in justification of the linearization principle for the nonlinear problem.

MSC:

35L60 First-order nonlinear hyperbolic equations
35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35L50 Initial-boundary value problems for first-order hyperbolic systems
80A30 Chemical kinetics in thermodynamics and heat transfer
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