Blow-up results of the positive solution for a weakly coupled quasilinear parabolic system

<p style='text-indent:20px;'>The main purpose of the present paper is to study the blow-up problem of a weakly coupled quasilinear parabolic system as follows:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left\{ \begin{array}{ll} u_{t} = \nabla\cdot\left(r(u)\nabla u\right)+f(u,v,x,t), &amp; \\ v_{t} = \nabla\cdot\left(s(v)\nabla v\right)+g(u,v,x,t) &amp;{\rm in} \ \Omega\times(0,t^{*}), \\ \frac{\partial u}{\partial\nu} = h(u), \ \frac{\partial v}{\partial\nu} = k(v) &amp;{\rm on} \ \partial\Omega\times(0,t^{*}), \\ u(x,0) = u_{0}(x), \ v(x,0) = v_{0}(x) &amp;{\rm in} \ \overline{\Omega}. \end{array} \right. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>Here <inline-formula><tex-math id="M1">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a spatial bounded region in <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^{n} \ (n\geq2) $\end{document}</tex-math></inline-formula> and the boundary <inline-formula><tex-math id="M3">\begin{document}$ \partial\Omega $\end{document}</tex-math></inline-formula> of the spatial region <inline-formula><tex-math id="M4">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is smooth. We give a sufficient condition to guarantee that the positive solution <inline-formula><tex-math id="M5">\begin{document}$ (u,v) $\end{document}</tex-math></inline-formula> of the above problem must be a blow-up solution with a finite blow-up time <inline-formula><tex-math id="M6">\begin{document}$ t^* $\end{document}</tex-math></inline-formula>. In addition, an upper bound on <inline-formula><tex-math id="M7">\begin{document}$ t^* $\end{document}</tex-math></inline-formula> and an upper estimate of the blow-up rate on <inline-formula><tex-math id="M8">\begin{document}$ (u,v) $\end{document}</tex-math></inline-formula> are obtained.</p>

INVERSE PROBLEMS FOR MATHEMATICAL MODELS WITH THE POINTWISE OVERDETERMINATION

In the article we examine well-posedness questions in the Sobolev spaces of an inverse source problem in the case of a quasilinear parabolic system of the second order. These problem arise when describing heat and mass transfer, diffusion, filtration, and in many other fields. The main part of the operator is linear. The unknowns occur in the nonlinear right-hand side. In particular, this class of problems includes the coefficient inverse problems on determinations of the lower order coefficients in a parabolic equation or a system. The overdetermination conditions are the values of a solution at some collection of points lying inside the spacial domain. The Dirichlet and oblique derivative problems under consideration. The problems are studied in a bounded domain with smooth boundary. However, the results can be generalized to the case of unbounded domains as well for which the corresponding solvability theorems hold. The conditions ensuring local (in time) well-posedness of the problem in the Sobolev classes are exposed. The conditions on the data are minimal. The results are sharp. The problem is reduced to an operator equation whose solvability is proven with the use of a priori bounds and the fixed point theorem. A solution possesses all generalize derivatives occurring in the system which belong to the space with and some additional necessary smoothness in some neighborhood about the overdetermination points.

Well-posedness for coupled bulk-interface diffusion with mixed boundary conditions

In this paper, we consider a quasilinear parabolic system of equations describing coupled bulk and interface diffusion, including mixed boundary conditions. The setting naturally includes non-smooth domains Ω. We show local well-posedness using maximal

Blow-Up and Global Existence for a Quasilinear Parabolic System

The problem of solutions to a class of quasilinear coupling parabolic system was studied. By constructing weak upper-solutions and weak lower-solutions, we obtain the global existence and blow-up of solutions under appropriate conditions.

THE EXISTENCE AND ASYMPTOTIC BEHAVIOR OF PERIODIC SOLUTIONS TO A QUASILINEAR PARABOLIC SYSTEM

In this paper, we investigate the existence, stability and global attractivity of T-periodic solutions for a class of quasilinear parabolic equations under Robin boundary conditions. We obtain that periodic solutions exist if the inter-specific competition rates are weak. The numerical simulations are also presented to illustrate our result.