On fractional \(p\)-Laplacian parabolic problem with general data
The authors prove the existence of a weak solution that belongs to a suitable fractional Sobolev space for the considered problem \[ \begin{cases} u_t+(-\Delta^s_p)u=f(x,t)&\text{in }\Omega_T\equiv\Omega\times(0,T),\\ u=0&\text{in }(\mathbb R^N\setminus\Omega)\times(0,T),\\ u(x,0)=u(x)&\text{in }\Omega, \end{cases} \tag{P} \] in the case of \(p\neq 2\), by using the way limit of approximations. Moreover, if the data \((f,u_0)\) are nonnegative then they obtained the weak solution of problem (P) as an entropy nonnegative solution. Finally they also showed extinction in the finite time, nonfinite speed of propogation and extinction for concave case to the considered problem according the values of \(p\).
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