In this paper, we consider a nonlocal diffusion equation involving the fractional $p\left(x\right)$-Laplacian with nonlinearities of variable exponent type. Employing the subdifferential approach we establish the existence of local solutions. By combining the potential well theory with the Nehari manifold, we obtain the existence of global solutions and finite time blow-up of solutions. Moreover, we study the asymptotic stability of global solutions as time goes to infinity in some variable exponent Lebesgue spaces.

中文翻译：

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涉及分数 p(x)-拉普拉斯算子的抛物方程解的全局存在性和爆炸性解

在本文中，我们考虑一个涉及分数的非局部扩散方程$p\left(X\right)$-具有可变指数类型非线性的拉普拉斯算子。采用次微分方法，我们建立了局部解决方案的存在。通过将势阱理论与Nehari流形相结合，我们得到了全局解的存在性和解的有限时间爆炸。此外，我们研究了在一些可变指数 Lebesgue 空间中随着时间趋于无穷大的全局解的渐近稳定性。