On the Cauchy problem of evolution \(p\)-Laplacian equations with strongly nonlinear sources when \(1<p<2\) (Q5946545)

The solvability of the Cauchy problem for \[ u_t=\text{div}(|\text{D}u|^{p-2}\text{D}u)+{{u^q}\over{(1+|x|)^{\alpha}}}\quad\text{in } S_T=\mathbb{R}^{N}\times (0,T), \] \[ u(x,t)=u_0(x)\geq 0 \] is studied. Optimal growth conditions on the initial condition \(u_0\) are found for this Cauchy problem to have a local solution. Some sufficient conditions for the existence and nonexistence of global solutions are also provided. The paper extends results from \textit{J. N. Zhao} [J. Differ. Equations 121, No. 2, 329-383 (1995; Zbl 0836.35081)].