2
$\begingroup$

I am dealing with diffusion-reaction equation with three species. I have $L^2$ bound of concentrations. Now I want $L^2$ bound of gradient of concentrations. Somehow if I get $L^4$ or $L^\infty$ bound Of $u_i$, then I can solve my problem. Please help me.

The model is the following system of PDEs for the vector variable ($u_1, u_2, u_3$) $$ \begin{cases} \partial_t u_1- D_1 \Delta u_1=-k_fu_1u_2+k_b u_3+\theta_1 \\ \partial_t u_2- D_2 \Delta u_2=-k_fu_1u_2+k_b u_3+\theta_2 \\ \partial_t u_3- D_3 \Delta u_3=k_fu_1u_2-k_b u_3+\theta_3 \end{cases} $$ where

  • $D_i(i=1,2,3), k_b, k_f$ are positive constants and
  • $\theta_i(x,t)$ is a function in $L^2$.

Note.- I don't have positive solution of diffusion-reaction system.

Improve this question
$\endgroup$
4
  • 3
    $\begingroup$ It's difficult to help if you don't state the problem explicitly... I don't think there's a general recipe to follow. $\endgroup$
    – Beni Bogosel
    Commented Mar 31, 2023 at 7:26
  • 2
    $\begingroup$ This question should include more details and clarify the problem. $\endgroup$
    – Pietro Majer
    Commented Mar 31, 2023 at 7:39
  • $\begingroup$ What exactly do you mean by $L^2$ (or other $L^p$ bounds)? (space for positive times, mixed time-space,...) $\endgroup$
    – fedja
    Commented Mar 31, 2023 at 21:48
  • $\begingroup$ @fedja $L^2$ bound means $||u||_{L^2(0,T; \Omega)}$ and $||\nabla u||_{L^2(0,T; \Omega)}$, $\Omega$ is domain. $\endgroup$
    – Arghya kundu
    Commented Apr 1, 2023 at 9:56

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .