Global existence of solutions for a strongly coupled quasilinear parabolic system

We are interested in existence, uniqueness and a priori estimates of non-negative solutions to the following problem: $$ \begin{gathered} {u_{t}} = \Delta \left[ {\left( {1 + \alpha v} \right)u} \right] + uf(u,v), \hfill \\ {v_{t}} = \Delta v + vg(u,v)\quad in\;(0,\infty ) \times \Omega , \hfill \\ u = v = 0\quad in\;(0,\infty ) \times \partial \Omega , \hfill \\ u(0) = {u_{{0,}}}v(0) = {v_{0}}\quad in\;\Omega \hfill \\ \end{gathered} $$ (1.0) .

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