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Given that $$\frac{\partial u}{\partial t}+\sin(y)\frac{\partial u}{\partial x}=

u\Bigl(\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}\Bigr)$$ With the following periodic boundary conditions: $$u(-\pi,y,t)=u(\pi,y,t) \\ u(x,-\pi,t)=u(x,\pi,t) \\u_x(-\pi,y,t)=u_x(\pi,y,t)\\ u_y(x,-\pi,t)=u_y(x,\pi,t)\\ u(x,y,0)=F(x,y)$$ Prove that $$\lVert u \rVert_{L^2} \leq Ce^{-

u t}$$

I have used the finite Fourier transform to get that $$\frac{du_{mn}}{dt}=-

u (n^2+m^2)u_{mn} -\int_{-\pi}^{\pi}\sin(y) u_ne^{-imy}dy$$

Where $$u_{mn}=\int_{-\pi}^{\pi} \int_{-\pi}^{\pi} u(x,y,t)e^{-imx} *e^{-iny} dxdy$$

Second I tried Energy method Multiply by u and then integrate, still I didn't get the required result.

How to get the required result ? Any Hint ?