# Why there exist a solution when Evans constructs a parabolic equation in proving strong maximum principle

Why there exist a solution when Evans constructs a parabolic equation in proving the strong maximum principle theorem 11 in chapter 7.

The theorem 11 is under below .

Theorem 11 (Strong maximum principle ) . Assume $$u\in C_1^2(U_T)\cap (\overline{U}_T))$$ and $$c\equiv 0 \text{ in} U_T$$ Suppose also $$U$$ is connected.$$\cdots$$

Proof. 1. Assume $$u_t+Lu\leq 0 \text{in} U_T$$ and $$u$$ attains its maximum at some point $$(x_0,t_0)\in U_T$$ . Select a smooth , open set $$W\subset\subset U$$, with $$x_0\in W$$ . Let $$v$$ solve \left\{ \begin{align} v_t+Lv&=&0 \ \text{in} W_T \\ v&=&u \ \text{on} \ \Delta _T \end{align} \right. where $$\Delta _T$$ denotes the parabolic boundary of $$W_T$$ $$\cdots$$

My doubts :why there is a solution $$v$$? Existence of weak solution can guarantee if the condiction $$v=0$$ on $$\partial U×[0,1]$$. Could someone give me some advice, thank you!

The essential point is what Evans means by "parabolic boundary." It is not understood as the set $$\partial U \times [0,T]$$ as you write. Instead, if you check how this is defined in the book you'll see that $$W_T = W \times (0,T] \text{ and } \Delta_T = \overline{W_T} \backslash W_T = (W \times \{0\}) \cup (\partial W \times [0,T]).$$ The solvability theory for the PDE with data specified on $$\Delta_T$$ is exactly what Evans establishes in the material prior to Theorem 11 in Section 7.1, though there he specializes to the case that the data on $$\partial W \times [0,T]$$ is zero. To recover the general case, though, we can simply subtract $$u$$. In other words we first solve $$\begin{cases} w_t + L w = -u_t - L u & \text{in } W_T \\ w =0 & \text{on } \Delta_T \end{cases}$$ using the theory in Section 7.1. We then define $$v = w + u$$, which then solves
$$\begin{cases} v_t + L v = 0 & \text{in } W_T \\ v =u & \text{on } \Delta_T. \end{cases}$$