When I proved some lemma I don't know whether it is true or not because i want to use it in other theorem.
Consider the following IBVP with Dirichlet BCs on a bounded open set $Ω ⊂ R^n$ for $u : Ω × [0, T]→ R:$ $$u_t = ∆u + f(x, t) \quad x\in \Omega, \quad t \in[0,T]\\ u(x, t) = 0 \quad x \in ∂Ω,\quad t \in[0,T]\\ u(x, 0) = g(x) \quad x \in Ω.$$ We have the weak formulate of the above Equations $$(u_t(t), v)_{L_2} + a (u(t),v) = (f(t), v)_{L_2}\quad \text{ for } 0 \leq t \leq T$$ for every $v\in H^1_0(\Omega)$
My question is this equation true for all $t\in[0,T]$ because I find other source whose previous equation is true only almost all $t\in [0,T]$
precisely can I write the following equality for any $t\in[0,T]$ not for almost any $t\in[0,T]$ $$\int_{0}^{t}(u_s(s),\phi)ds+\int_{0}^{t}a(u(s),\phi)ds=\int_{0}^{t}(f(s),\phi)ds$$ then for an other function $v$ where $$\int_{0}^{t}(v_s(s),\phi)ds+\int_{0}^{t}a(v(s),\phi)ds\leq\int_{0}^{t}(f(s),\phi)ds$$ we have $$\int_{0}^{t}(v_s(s)-u_s(s),\phi)ds+\int_{0}^{t}a(v(s)-u(s),\phi)ds\leq 0$$
Repeat my question is this equation true for all $t\in[0,T]$ not for almost any $t\in[0,T]$ ?
As an additional when I apply gronwal inegality I find $y(t)=\|u(t,.)\|_{L^2(\Omega)}=0$ I need that for everything $t\in[0,T]$, especially for $t=0$ and $t=T$, i.e. $y(T)=0 \text{ and y(0)=0}$