We consider a periodic parabolic problem involving singular nonlinearity and homogeneous Dirichlet boundary condition modeled by$$\begin{aligned} \dfrac{\partial u}{\partial t}-\Delta u =\dfrac{f}{u^{\gamma }} \text { in }Q_{T}, \end{aligned}$$where \(T>0\) is a period, \(\Omega \) is an open regular bounded subset of \(\mathbb {R}^{N}\), \(Q_{T}=]0,T[\times \Omega \),\(\gamma \in \mathbb {R}\) and f is a nonnegative integrable function periodic in time with period T. Under a suitable assumptions on f, we establish the existence of a weak T-periodic solution for all ranges of value of \(\gamma \).

中文翻译：

---

具有奇异非线性和$$ L ^ {1} $$ L 1数据的半线性抛物问题的弱周期解

我们考虑一个周期抛物线问题，该问题涉及奇异非线性和由$$ \ begin {aligned} \ dfrac {\ partial u} {\ partial t}-\ Delta u = \ dfrac {f} {u ^ {\ gamma}} \ text {in} Q_ {T}，\ end {aligned} $$其中\（T> 0 \）是一个周期，\（\ Omega \）是\（\ mathbb { R} ^ {N} \），\（Q_ {T} =] 0，T [\ times \ Omega \），\（\ gamma \ in \ mathbb {R} \）和f是周期内的一个非负可积函数时间为T的时间。在f的适当假设下，我们建立了t的所有取值范围的弱T周期解的存在\（\ gamma \）。