I have this Cauchy-Dirichlet problem and i have to prove that the solution is non-negative in $\mathbb{R}^{+}\times\left[0,1\right]$. How can i do?
\begin{cases} u_t(t,x) - u_{xx} (t,x)= tx \quad\quad \mathbb{R}^{+}\times\left[0,1\right] \\ u(0,x) = \sin(\pi x) \quad\quad x\in\left[0,1\right]\\ u(t,0) = 2te^{1-t} \quad\quad t\in\left(0,+\infty\right)\\ u(t,1) = 1-\cos(\pi t) \quad\quad t\in\left(0,+\infty\right) \end{cases}
