In this paper we study a class of nonlinear quasi-linear diffusion equations involving the fractional \(p(\cdot )\)-Laplacian with variable exponents, which is a fractional version of the nonhomogeneous \(p(\cdot )\)-Laplace operator. The paper is divided into two parts. In the first part, under suitable conditions on the nonlinearity f, we analyze the problem \(({\mathscr {P}}_{1})\) in a bounded domain \(\varOmega \) of \({\mathbb {R}}^N\) and we establish the well-posedness of solutions by using techniques of monotone operators. We also study the large-time behaviour and extinction of solutions and we prove that the fractional \(p(\cdot )\)-Laplacian operator generates a (nonlinear) submarkovian semigroup on \(L^{2}(\varOmega ).\) In the second part of the paper we establish the existence of global attractors for problem \(({\mathscr {P}}_{2})\) under certain conditions in the potential \({\mathbb {V}}.\) Our results are new in the literature, both for the case of variable exponents and for the fractional p-laplacian case with constant exponent.

中文翻译：

---

包含分数$$ p（\ cdot）$$ p（·）-Laplacian的非局部扩散方程

在本文中，我们研究了一类涉及分数\（p（\ cdot）\）-具有可变指数的拉普拉斯算子的非线性拟线性扩散方程，这是非齐次\（p（\ cdot）\）的分数形式。拉普拉斯运算符。本文分为两部分。在第一部分中，在非线性f的合适条件下，我们分析\（{\ mathbb的有界域\（\ varOmega \）中的问题\（（{{mathscr {P}} _ {1}）\）{R}} ^ N \），然后我们通过使用单调算符的技术来确定解的适定性。我们还研究了溶液的长时间行为和灭绝，并证明了分数\（p（\ cdot）\）-Laplacian算子在\（L ^ {2}（\ varOmega）。\）上生成一个（非线性）子马尔可夫半群。在本文的第二部分中，我们建立了问题\（（{{mathscr {P} } _ {2}）\）在一定条件下可能的\（{\ mathbb {V}}。\）。对于可变指数的情况和分数p-laplacian情况，我们的结果在文献中都是新的。常数指数。