Consider the following Cauchy problem $$u_t - \Delta u + f(x,u,u_x) = 0, \quad (t,x) \in (0,T) \times \mathbb{R}^n,$$ with initial condition $u(0,x) = g(x), \ x \in \mathbb{R}^n.$

Suppose that $u \in C^2((0,T) \times \mathbb{R}^n)$ is a solution of the initial value problem.

Can we obtain (under some reasonable assumptions) that $$u_t(0,x) - \Delta u(0,x) + f(x,u,u_x)= 0 \ ?$$

Consider the following Cauchy problem $$u_t - \Delta u + f(x,u,u_x) = 0, \quad (t,x) \in (0,T) \times \mathbb{R}^n,$$ with initial condition $u(0,x) = g(x), \ x \in \mathbb{R}^n.$

Suppose that $u \in C^2((0,T) \times \mathbb{R}^n)$ is a solution of the initial value problem.

Can we obtain (under some reasonable assumptions) that $$u_t(0,x) - \Delta u(0,x) + f(x,u,u_x)= 0,$$ that is, that the solution $u$ solves the PDE at time $t=0$ too?

Consider the following Cauchy problem $$u_t - \Delta u + f(x,u,u_x) = 0, \quad (t,x) \in (0,T) \times \mathbb{R}^n,$$ with initial condition $u(0,x) = g(x), \ x \in \mathbb{R}^n.$

Suppose that $u \in C^2((0,T) \times \mathbb{R}^n)$ is a solution of the IVP.

Can we obtain (under some reasonable assumptions) that $$u_t(0,x) - \Delta u(0,x) + f(x,u,u_x)= 0 \ ?$$

Consider the following Cauchy problem $$u_t - \Delta u + f(x,u,u_x) = 0, \quad (t,x) \in (0,T) \times \mathbb{R}^n,$$ with initial condition $u(0,x) = g(x), \ x \in \mathbb{R}^n.$

Suppose that $u \in C^2((0,T) \times \mathbb{R}^n)$ is a solution of the initial value problem.

Can we obtain (under some reasonable assumptions) that $$u_t(0,x) - \Delta u(0,x) + f(x,u,u_x)= 0 \ ?$$