# Continuity of solution of a parabolic PDE w.r.t. system parameters

If we have a system of PDE of the form:

$$\begin{cases} \dfrac{\partial y}{\partial t}(t,x)=D\Delta y+F(t,x,f(x),y) ,\ (t,x)\in (0,T)\times\Omega\\ \dfrac{\partial y}{\partial \nu}(t,x)=0,\ (t,x)\in (0,T)\times\partial\Omega \\ y(0,x)=y_{0}(x),\ x\in\Omega\end{cases}$$

with a unique solution $$y=y^f$$(mild solution). All derivatives are considered in the distributional sense (weak derivatives).Here $$\Omega\subset\mathbb{R}^N$$ is an open and bounded set with $$C^1$$ boundary, $$y=(y_1,y_2,...,y_m)$$, $$F=(F_1,F_2,\dots,F_m)$$, $$D=diag(d_1,d_2,\dots, d_m),\ d_i\in\mathbb{R},\ \forall\ i=1,2,...,m$$ and $$f:\Omega\to\mathbb{R}$$ a function in $$L^1(\Omega)$$. We consider the following approximation of the system:

$$\begin{cases} \dfrac{\partial y}{\partial t}(t,x)=D\Delta y+F(t,x,f_n(x),y) ,\ (t,x)\in (0,T)\times\Omega\\ \dfrac{\partial y}{\partial \nu}(t,x)=0,\ (t,x)\in (0,T)\times\partial\Omega \\ y(0,x)=y_{0}(x),\ x\in\Omega\end{cases}$$,

with the unique solution $$y^{f_n}$$.

How can we prove that if $$f_n\to f$$ in $$L^1(\Omega)$$, then $$y_i^{f_n}\to y_i^f$$ in $$L^p((0,T)\times\Omega),\ \forall\ i=1,2,\dots,n$$, for some $$p\in [1,\infty)$$? Are there some results in that sense? Maybe with a more general setting...?

Are there some estimates?

$$||y^{f_n}-y^f||\leq C||f_n-f||_{L^1(\Omega)}$$.

• What are conditions on $F$? – Andrew Jan 30 at 18:26
• We can assume any type of conditions (Lipschitz-type probably). I only want a method or some references about that type of parabolic problems. – Bogdan Jan 30 at 18:47
• Do you have the same question for an equation rather for a system? – Giorgio Metafune Jan 31 at 10:26
• It will be welcomed. – Bogdan Jan 31 at 10:51

This is only a sketch of an argument that can be used. Assume that $$F$$ is Lipscthitz and let $$y_1,y_2$$ be the solutions corresponding to $$f_1, f_2$$. If $$v=y_2-y_1$$, then $$|v_t-\Delta v|=|F(f_2,y_2)-F(f_2,y_1)+F(f_2, y_1)-F(f_1,y_1)| \le L(|v|+|f_2-f_1|)$$ with zero bc and initial value. If $$T(t)$$ is the semigroup generated by $$\Delta$$ with the appropriate bc, then $$|v(t)| \le L\int_0^t T(t-s)|v(s)|ds +L\int_0^t T(t-s)(|f_2-f_1|)ds.$$ Next use the fact that $$T(t)$$ is contractive in $$L^p$$ and maps $$L^1$$ to $$L^p$$ with norm less that $$Ct^{-N/p'}$$. If $$t \le \delta$$, taking $$L^p$$ norms (in space) we get $$\|v(t)\|_p \le L\delta \sup_{0 \le t \le \delta}\|v(t)\|_p+CL\|f_2-f_1\|_1\int_0^\delta t^{-N/p'}dt.$$ Now take $$p so that $$t^{-N/p'}$$ is integrable near $$0$$ and take the supremum of the left hand side over $$[0, \delta]$$. If $$L\delta \le 1/2$$ this gives $$\sup_{0 \le t \le \delta} \|v(t)\|\le C_1 \|f_2-f_1\|_1.$$ Now the argument can be iterated to larger intervals.