*3.2. Weak Formulation*

Rewriting the (3) in unified form, we get

$$\begin{cases} \partial\_t u + \partial\_t^{1-\kappa} A u = f(x, t) + \sum\_{1 \le i \le P} I\_i(u(\cdot, t\_i)) \delta(t - t\_i) & \text{in} \quad \Omega \times (0, T), \\ u = 0 & \text{on} \quad \Gamma \times (0, T), \\ u(\cdot, 0) = u\_0 & \text{in} \quad \Omega. \end{cases} \tag{16}$$

A weak formulation of (16) is to find a *<sup>u</sup>* <sup>∈</sup> *PC*(0, *<sup>T</sup>*; *<sup>H</sup>*<sup>1</sup> <sup>0</sup> (Ω)) such that

$$(\partial\_t u, \upsilon) + (\partial\_t^{1-a} A u, \upsilon) = (f, \upsilon) + \sum\_i (I\_i(u(\cdot, t\_i)) \delta(t - t\_i), \upsilon), \upsilon \in H\_0^1(\Omega). \tag{17}$$

Thus, we have a variational form of (16) as follows:

$$(\partial\_t u, v)dt + a(u, v) = l(v),\tag{18}$$

where,

$$a(\boldsymbol{u}, \boldsymbol{v}) = (\partial\_t^{1-a} A \boldsymbol{u}, \boldsymbol{v}) = \int\_{\Omega} \partial\_t^{1-a} \nabla \boldsymbol{u} \cdot \nabla \boldsymbol{v} d\boldsymbol{x},$$

$$l(\boldsymbol{v}) = (f, \boldsymbol{v}) dt + \sum\_{i=1}^{P} (I\_i(\boldsymbol{u}(t\_i)) \delta(t - t\_i), \boldsymbol{v})\_i$$

with the following conditions:


**Definition 1.** *A function u* : [0, *<sup>T</sup>*] <sup>→</sup> *<sup>H</sup>*<sup>1</sup> <sup>0</sup> (Ω) *is called a weak solution of* (3) *if :*


Based on the above analysis, we can now formulate the following two theorems.

**Theorem 1.** *For every <sup>f</sup>* <sup>∈</sup> *<sup>L</sup>*2(0, *<sup>T</sup>*; *<sup>H</sup>*−1(Ω)) *and <sup>u</sup>*<sup>0</sup> <sup>∈</sup> *<sup>H</sup>*<sup>1</sup> <sup>0</sup> (Ω), *there exists a unique weak solution <sup>u</sup>* <sup>∈</sup> *<sup>L</sup>*2(0, *<sup>T</sup>*; *<sup>H</sup>*<sup>1</sup> <sup>0</sup> (Ω)) <sup>∩</sup> *PC*(0, *<sup>T</sup>*; *<sup>H</sup>*<sup>1</sup> <sup>0</sup> (Ω)) *of* (3)*.*

**Proof.** Existence and uniqueness of weak solution is followed by the Lax-Milgram theorem.

**Theorem 2.** *For every <sup>f</sup>* <sup>∈</sup> *<sup>L</sup>*2(0, *<sup>T</sup>*; *<sup>H</sup>*−1(Ω)) *and <sup>u</sup>*<sup>0</sup> <sup>∈</sup> *<sup>H</sup>*<sup>1</sup> <sup>0</sup> (Ω), *there exists a unique mild solution <sup>u</sup>* <sup>∈</sup> *<sup>L</sup>*2(0, *<sup>T</sup>*; *<sup>H</sup>*<sup>1</sup> <sup>0</sup> (Ω)) <sup>∩</sup> *PC*(0, *<sup>T</sup>*; *<sup>H</sup>*<sup>1</sup> <sup>0</sup> (Ω)) *of* (3) *and given by* (15)*.*
