Consider the following approximate problem:
$$\begin{aligned}& u_{\varepsilon t} - \varepsilon\operatorname{div}\bigl(|\nabla u_{\varepsilon }|^{p_{+}-2}\nabla u_{\varepsilon}\bigr) -\operatorname{div} \bigl( a(x) \vert \nabla{u_{\varepsilon}} \vert ^{p(x)- 2}\nabla u_{\varepsilon}\bigr) - \vec{b}(x)\cdot\nabla u_{\varepsilon}^{q} \\& \quad = 0,\quad (x,t)\in{Q_{T}}, \end{aligned}$$
(2.1)
$$\begin{aligned}& {u_{\varepsilon}}(x,t) = 0, \quad (x,t) \in\partial\Omega \times (0,T), \end{aligned}$$
(2.2)
$$\begin{aligned}& {u_{\varepsilon}}(x,0) = {u_{\varepsilon,0}}(x), \quad x\in\Omega , \end{aligned}$$
(2.3)
where \(0\leq u_{\varepsilon,0} \in C^{\infty}_{0}(\Omega)\), \(|u_{\varepsilon ,0}|_{L^{\infty}(\Omega)}\leq|u_{0}|_{L^{\infty}(\Omega)}\), \(a(x) \vert \nabla u_{\varepsilon,0} \vert ^{p_{+}}\) uniformly is convergent to \(a(x)|\nabla u_{0}(x)|^{p_{+}}\) in \({L^{1}}(\Omega)\). Then there is an unique nonnegative solution \(u_{\varepsilon}\in L^{p_{+}}(0,T; W^{1,p_{+}}_{0}(\Omega))\) [6].
By the maximum principle ([8], p. 150), we have
$$ \Vert u_{\varepsilon} \Vert _{L^{\infty}(Q_{T})} \leqslant c. $$
(2.4)
Since
$$\begin{aligned} \iint_{{Q_{T}}} \bigl\vert u_{\varepsilon}\vec{b}(x)\cdot\nabla u_{\varepsilon }^{q} \bigr\vert \,dx\,dt =&q \iint_{{Q_{T}}} \bigl\vert u^{q}_{\varepsilon} \vec{b}(x)\cdot\nabla u_{\varepsilon} \bigr\vert \\ \leq&\frac{\varepsilon}{2} \iint_{{Q_{T}}} \vert \nabla u_{\varepsilon } \vert ^{p_{+}}\,dx\,dt+c(\varepsilon), \end{aligned}$$
by multiplying (2.1) with \(u_{\varepsilon}\) and integrating it over \(Q_{T}\), we easily obtain
$$ \frac{1}{2} \int_{\Omega}u_{\varepsilon}^{2}\,dx+\varepsilon \iint _{{Q_{T}}}|\nabla u_{\varepsilon}|^{p^{+}}\,dx\,dt + \iint_{{Q_{T}}} a(x)|\nabla u_{\varepsilon}|^{p(x)}\,dx\,dt \leqslant c. $$
(2.5)
For any \(\Omega_{\lambda}\subset\subset\Omega\), since \(p_{-}=\min_{x\in\overline{\Omega} }p(x)>1\), by (2.5),
$$ \int_{0}^{T} \int_{\Omega_{\lambda}}|\nabla u_{\varepsilon }|\,dx\,dt\leq c \biggl( \int_{0}^{T} \int_{\Omega_{\lambda}}|\nabla u_{\varepsilon}|^{p_{-}}\,dx\,dt \biggr)^{\frac{1}{p_{-}}}\leq c(\lambda ) $$
(2.6)
and
$$ \varepsilon \iint_{Q_{T}}|\nabla u_{\varepsilon}|^{p_{+}}\,dx\,dt\leq c. $$
(2.7)
For any \(v\in L^{p_{+}}(0,T; W^{1,p_{+}}_{0}(\Omega))\), \(\|v\| _{L^{p_{+}}(0,T; W^{1,p_{+}}_{0}(\Omega))}=1\). We have
$$\begin{aligned}& \langle u_{\varepsilon t}, v\rangle+\varepsilon \iint_{Q_{T}}|\nabla u_{\varepsilon }|^{p_{+}-2}\nabla u_{\varepsilon}\nabla v\,dx\,dt \\& \quad {}+ \iint_{Q_{T}}a(x)|\nabla\nu_{\varepsilon}|^{p(x)-2}\nabla u_{\varepsilon}\nabla v\,dx\,dt \\& \quad {}+ \iint_{Q_{T}}u^{q} \bigl(b^{i}_{x_{i}}(x)v+b^{i}(x)v_{x_{i}} \bigr)\,dx\,dt. \end{aligned}$$
(2.8)
This formula makes sense because \(u_{\varepsilon}\in L^{p_{+}}(0,T; W^{1,p_{+}}_{0}(\Omega))\cap L^{\infty}(Q_{T})\).
By (2.8), using the Young inequality, we can show that
$$\begin{aligned} \bigl\vert \langle u_{\varepsilon t}, v\rangle \bigr\vert \leq& c \biggl[ \varepsilon \iint_{Q_{T}} \vert \nabla u_{\varepsilon} \vert ^{p_{+}}\,dx\,dt+ \iint_{{Q_{T}}} a(x) \vert \nabla u_{\varepsilon} \vert ^{p(x)}\,dx\,dt \\ &{}+ \iint_{{Q_{T}}} \bigl( \vert v \vert ^{p_{+}}+ \vert \nabla v \vert ^{p_{+}}\bigr)\,dx\,dt+1 \biggr]\leq c, \end{aligned}$$
then
$$ \| u_{\varepsilon t}\|_{L^{p'_{+}}(0,T;W^{-1,p'_{+}}(\Omega))}\leq c, $$
(2.9)
where \(p'_{+}=\frac{p_{+}}{p_{+}-1}\).
Now, for any \(\varphi\in C_{0}^{1}(\Omega)\), \(0\leq\varphi\leq1\), for any \(v\in L^{p_{+}}(0,T; W^{1,p_{+}}_{0}(\Omega))\),
$$\begin{aligned} \bigl\langle (\varphi u_{\varepsilon})_{t}, v\bigr\rangle =&\langle \varphi u_{\varepsilon t}, v\rangle =\langle u_{\varepsilon t}, \varphi v\rangle \\ =&\varepsilon \iint_{Q_{T}} \vert \nabla u_{\varepsilon} \vert ^{p_{+}-2}\nabla u_{\varepsilon}\nabla(\varphi v)\,dx\,dt \\ &{}+ \iint_{Q_{T}}a(x) \vert \nabla u_{\varepsilon} \vert ^{p(x)-2}\nabla u_{\varepsilon}\nabla(\varphi v) \\ &{}+ \iint_{Q_{T}}u^{q} \bigl(b^{i}_{x_{i}}(x)v \varphi+b^{i}(x) (v\varphi )_{x_{i}}\bigr)\,dx \,dt. \end{aligned}$$
(2.10)
Similarly, we can show that
$$\begin{aligned} \bigl\vert \bigl\langle (\varphi u_{\varepsilon})_{t}, v\bigr\rangle \bigr\vert =&\langle u_{\varepsilon t}, \varphi v\rangle \\ \leq& c \biggl[\varepsilon \iint_{Q_{T}} \vert \nabla u_{\varepsilon} \vert ^{p_{+}}\,dx\,dt+ \iint_{{Q_{T}}} a(x) \vert \nabla u_{\varepsilon} \vert ^{p(x)}\,dx\,dt \\ &{}+ \iint_{{Q_{T}}} \bigl( \vert v \vert ^{p_{+}}+ \vert \nabla v \vert ^{p_{+}}\bigr)\,dx\,dt+1 \biggr]\leq c, \end{aligned}$$
then
$$ \bigl\Vert (\varphi u_{\varepsilon})_{t} \bigr\Vert _{L^{p'_{+}}(0,T;W^{-1,{p'_{+}}}(\Omega))}\leq c. $$
(2.11)
For a fixed s such that \(s>\frac{N}{2}+1\), one has \(H_{0}^{s}(\Omega )\hookrightarrow W^{1, p_{+}}(\Omega)\). Consequently, \(W^{-1,{p'_{+}}}(\Omega)\hookrightarrow H^{-s}(\Omega)\). As a result, we have
$$ \bigl\Vert (\varphi u_{\varepsilon})_{t} \bigr\Vert _{L^{p'_{+}}(0,T;H^{-s}(\Omega))}\leq c. $$
(2.12)
At the same time, we have
$$\iint_{Q_{T}} \bigl\vert \nabla(\varphi u_{\varepsilon}) \bigr\vert ^{p_{-}}\,dx\,dt\leq c(\varphi) \biggl(1+ \int_{0}^{T} \int_{\Omega_{\varphi}} \vert \nabla u_{\varepsilon} \vert ^{p_{-}} \,dx\,dt\biggr)\leq c(\varphi), $$
where \(\Omega_{\varphi}=\operatorname{supp}\varphi\). We give an explanation of this inequality. Since \(a(x)\in C^{1}(\overline{\Omega})\), \(a(x)>0\) when \(x\in\Omega\), for any \(\varphi(x)\in C_{0}^{1}(\Omega)\),
$$\frac{\varphi(x)}{a(x)}\leq c(\varphi), $$
then
$$\begin{aligned}& \int_{0}^{T} \int_{\Omega_{\varphi}}|\nabla u_{\varepsilon }|^{p_{-}} \,dx\,dt \\& \quad = \int_{0}^{T} \int_{\Omega}\varphi(x)|\nabla u_{\varepsilon}|^{p_{-}}\,dx \,dt \\& \quad = \int_{0}^{T} \int_{\Omega}\frac{\varphi(x)}{a(x)}a(x)|\nabla u_{\varepsilon}|^{p_{-}} \,dx\,dt \\& \quad \leq c(\varphi) \int_{0}^{T} \int_{\Omega}a(x)|\nabla u_{\varepsilon }|^{p_{-}}\,dx\,dt \\& \quad \leq c(\varphi). \end{aligned}$$
Thus we have
$$ \|\varphi u_{\varepsilon}\|_{L^{p'_{+}}(0,T;W^{1,p_{-}}_{0}(\Omega ))}\leq c. $$
(2.13)
Noticing that \(W^{1,p_{-}}_{0}(\Omega)\hookrightarrow L^{p_{-}}(\Omega )\hookrightarrow H^{-s}(\Omega)\), we can employ Aubin’s compactness theorem in [12] to obtain \(\varphi u_{\varepsilon}\rightarrow\varphi u\) strongly in \(L^{p'_{+}}(0,T;L^{p_{-}}(\Omega))\). Thus \(\varphi u_{\varepsilon}\rightarrow\varphi u\) a.e. in \(Q_{T}\). In particular, due to the arbitrariness of φ, \(u_{\varepsilon }\rightarrow u\) a.e. in \(Q_{T}\).
Now, by (2.7),
$$\varepsilon|\nabla u_{\varepsilon}|^{p_{+}-2}\nabla u_{\varepsilon}\rightharpoonup0, \quad \text{in } L^{\frac{p_{+}}{p_{+}-1}}(Q_{T}). $$
By (2.4), (2.5), (2.6), there exists a function u and an n-dimensional vector function \(\overrightarrow{\zeta}= ({\zeta_{1}}, \ldots,{\zeta_{n}})\) satisfying
$$u \in L^{\infty}(Q_{T}),\qquad \vert \zeta_{i} \vert \in L^{\frac {p(x)}{p(x) - 1}}(Q_{T}), $$
and
$$\begin{aligned}& {u_{\varepsilon}} \rightharpoonup u,\quad \text{weakly-star in } {L^{\infty}(Q_{T})}, \qquad u_{\varepsilon }\rightarrow u, \quad \text{a.e. in } Q_{T}, \\& u_{\varepsilon}^{q}\rightarrow u^{q}, \quad \text{a.e. in } Q_{T}, \\& {u_{\varepsilon x_{i}}} \rightharpoonup u_{x_{i}} ,\quad \text{in } L_{\mathrm{loc}}^{p(x)}(Q_{T}), \\& a(x) \vert \nabla u_{\varepsilon} \vert ^{p(x)- 2} \nabla u_{\varepsilon} \rightharpoonup\overrightarrow{\zeta},\quad \text{in } \bigl\{ L^{1}\bigl(0,T; L^{\frac{p(x)}{p(x)-1}}(\Omega)\bigr)\bigr\} ^{N}. \end{aligned}$$
Moreover, we can prove that
$$ \iint_{Q_{T}} a \vert \nabla u \vert ^{p(x) - 2}\nabla u \cdot \nabla\varphi \,dx\,dt = \iint_{Q_{T}} \overrightarrow{\zeta} \cdot\nabla\varphi \,dx\,dt, $$
(2.14)
for any given function \(\varphi\in C_{0}^{1} ({Q_{T}})\).
Then, by (2.14), for any \(\varphi\in C_{0}^{1}(Q_{T})\), we have
$$ \langle u_{t},\varphi\rangle + \iint_{{Q_{T}}} \bigl[a(x) \vert \nabla u \vert ^{p(x)- 2} \nabla u \nabla \varphi+u^{q}\bigl(b^{i}(x)\varphi _{x_{i}}+b^{i}_{x_{i}}(x)\varphi\bigr) \bigr]\,dx\,dt = 0. $$
(2.15)
If for any given \(t\in[0, T)\), denoting \(\Omega_{\varphi }=supp\varphi\), then we have
$$ \langle u_{t},\varphi\rangle + \int_{0}^{T} \int_{\Omega_{\varphi}}\bigl[a(x) \vert \nabla u \vert ^{p(x)- 2} \nabla u \nabla \varphi +u^{q}\bigl(b^{i}(x) \varphi_{x_{i}}+b^{i}_{x_{i}}(x)\varphi\bigr) \bigr]\,dx \,dt = 0. $$
(2.16)
Moreover, for any \(\varphi_{1}\in C_{0}^{1} ({Q_{T}})\), \(\varphi_{2}(x,t)\in L^{\infty}(0,T; W^{1,p(x)}_{\mathrm{loc}}(\Omega))\), we clearly have
$$g(\varphi_{1}\varphi_{2})\in L^{\infty}\bigl(0,T; W_{0}^{1,p(x)}(\Omega _{\varphi_{1}})\bigr). $$
Since \(C_{0}^{\infty}(\Omega_{\varphi_{1}})\) is dense in \(W_{0}^{1, p(x)}(\Omega_{\varphi_{1}})\), by taking a limit, we have
$$\begin{aligned}& \bigl\langle u_{t},g(\varphi_{1}\varphi_{2})\bigr\rangle + \int_{0}^{T} \int_{\Omega_{\varphi_{1}}} \bigl[a(x) \vert \nabla u \vert ^{p(x)- 2} \nabla u \nabla g(\varphi_{1}\varphi_{2}) \bigr]\,dx\,dt \\& \qquad {}+ \int_{0}^{T} \int_{\Omega_{\varphi_{1}}} \bigl[u^{q} \bigl(b^{i}(x)g_{x_{i}}( \varphi_{1}\varphi_{2})+b^{i}_{x_{i}}g( \varphi_{1}\varphi _{2}) \bigr) \bigr]\,dx\,dt \\& \quad = 0, \end{aligned}$$
(2.17)
and so
$$\begin{aligned}& \bigl\langle u_{t},g(\varphi_{1}\varphi_{2})\bigr\rangle + \int_{0}^{T} \int_{\Omega} \bigl[a(x) \vert \nabla u \vert ^{p(x)- 2} \nabla u \nabla g(\varphi _{1}\varphi_{2}) \bigr]\,dx\,dt \\& \qquad {}+ \int_{0}^{T} \int_{\Omega} \bigl[u^{q} \bigl(b^{i}(x)g_{x_{i}}( \varphi _{1}\varphi_{2})+b^{i}_{x_{i}}g( \varphi_{1}\varphi_{2}) \bigr) \bigr]\,dx\,dt \\& \quad = 0. \end{aligned}$$
(2.18)
Finally, the initial value condition in the sense of (1.7) can be proved (1.7) as in [2], then u is a solution of Eq. (1.1) with the initial value (1.2) in the sense of Definition 1.3. Thus we have Theorem 1.4.