PDEs
Second-order parabolic equations generalize the heat equation:
where and initial time. Sobolev Space Valued Function .
Parabolic Differential Operator
Misplaced & -\sum_{i=1}^n \sum_{j=1}^n \partial_{x_j} \left( A^{ij}(t, x) \partial_{x_i} u \right) + \sum_{i=1}^n b^i(t, x) \partial_{x_i} u + c(t, x) u & \text{divergence form}\\ \\ -\sum_{i=1}^n \sum_{j=1}^n A^{ij}(t, x) \partial_{x_i x_j} u + \sum_{i=1}^n b^i(t, x) \partial_{x_i} u + c(t, x) u & \text{non-divergence form}\\ \end{cases}$$ if the matrix $A^{ij}(x)$ is uniformly **elliptic** $\forall x \in U, t \in [0, T]$: $$\left\langle y, A(t, x)y \right\rangle = \sum_{i=1}^n \sum_{j=1}^n A^{ij}(t, x) y_i y_j \geq \theta ||y||^2$$ for some $\theta > 0$ and $\forall \space y \in \mathbb{R}^n$. - $A^{ij}(t, \cdot)$ represents the (an)isoptropic diffusion of $u$ throughout $U$ *at time* $t$ - $b^i(t, \cdot)$ represents the transport of $u$ throughout $U$ *at time* $t$ - $c(t, \cdot)$ represents local increase or decrease of $u$ in $U$ *at time* $t$