This paper studies solving the Cauchy problem for singular \(p\)-Laplacian evolution equations. Specifically, for \(\mu\) a non-negative Radon measure, \( \frac{2N}{N+1}<p<2\), consider the following problem:
\[ \begin{aligned} u_t-\text{div}(|Du|^{p-2}Du)= \lambda |Du|^l+ \lambda_0u^q &\quad\text{in }\mathbb R^N\times(0,T),\\ u(x,0)=\mu &\quad\text{on }\mathbb R^N, \end{aligned} \]
with \(0<l<p,\) \(q\geq 1\) and \(\lambda,\lambda_0\geq 0\). The main theorem the authors obtain is that, if
\[ [\mu]:= \sup_{x\in\mathbb R^N}\;\sup_{0<\rho <1} \left(\rho^\theta\frac{1}{|B_\rho(x)|}\int_{B_\rho(x)}\,d\mu \right) <\infty , \]
with \(0\leq \theta \leq N\), \(\theta(l-p+1)<(p-l)\) and \(\theta(q-1)<p+\theta(p-2)\), then there exists a solution to the above initial value problem such that, for \(0<t<T_0\),
\[ [u]_t:= \sup_{0<\tau<t}\;\sup_{x\in\mathbb R^N}\;\sup_{0<\rho<1} \left(\frac{\rho^\theta}{|B_\rho(x)|}\int_{B_\rho(x)}u(y,\tau) \,dy\right)\leq \gamma \left([\mu]^{\frac{\rho}{\kappa}}+1\right) \]
and
\[ \|u(\cdot ,t)\|_{\infty,\mathbb R^N}\leq \gamma t^{-(\frac{\theta}{p+\theta (p-2)})} \left([\mu]^{\frac{\rho}{\kappa}}+1\right), \]
with \(T_0= T_0([\mu] ,N,p,q,\lambda, l,\theta,\lambda_0)\), \(\gamma=\gamma (N,p,q,\lambda ,l,\theta,\lambda_0)\) and \(\kappa =(p+N(p-2))\). The authors first establish a number of a priori estimates and then use these with an approximation method to prove the theorem. They conclude the paper by proving a second theorem for the supercritical case \(q>p-1+p/N\). The second theorem says for \(u\) a non-negative weak solution, for \(1<p<2\), and \(q>(1+\varepsilon)(p-1)+p/N\), there exist constants \(\rho_0\), \(0<\rho_0<1\), and \(\gamma \) so that
\[ \rho^{\frac{p}{q-(1+\varepsilon)(p-1)}} \left(\frac{1}{|B_\rho(x)|} \int_{B_\rho(x)} u(y,t)\,dy\right) \leq \gamma, \]
for all \(x\in\mathbb R^N\), \(0<t<T/2\), \(0<\varepsilon <p-1\) and \(0<\rho<\rho_0\).
They discuss in detail how their results relate to other recent results, and what extensions can be proved. Related work includes [D. Andreucci, Trans. Am. Math. Soc. 349, No. 10, 3911–3923 (1997; Zbl 0885.35056); D. Andreucci and E. Di Benedetto, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 18, No. 3, 363–441 (1991; Zbl 0762.35052); J. Zhao and P. D. Lei, Acta Math. Sin., Engl. Ser. 17, No. 3, 455–470 (2001; Zbl 0983.35071)].