Instantaneous shrinking and single point extinction for viscous Hamilton–Jacobi equations with fast diffusion

Abstract

For a large class of non-negative initial data, the solutions to the quasilinear viscous Hamilton–Jacobi equation $\partial _t u-\Delta _p u+|\nabla u|^q=0$ in $(0,\infty )\times \mathbb {R}^N$ are known to vanish identically after a finite time when $2N/(N+1)<p\le 2$ and $q\in (0,p-1)$. Further properties of this extinction phenomenon are established herein: instantaneous shrinking of the support is shown to take place if the initial condition $u_0$ decays sufficiently rapidly as $|x|\rightarrow \infty$, that is, for each $t>0$, the positivity set of u(t) is a bounded subset of $\mathbb {R}^N$ even if $u_0>0$ in $\mathbb {R}^N$. This decay condition on $u_0$ is also shown to be optimal by proving that the positivity set of any solution emanating from a positive initial condition decaying at a slower rate as $|x|\rightarrow \infty$ is the whole $\mathbb {R}^N$ for all times. The time evolution of the positivity set is also studied: on the one hand, it is included in a fixed ball for all times if it is initially bounded (localization). On the other hand, it converges to a single point at the extinction time for a class of radially symmetric initial data, a phenomenon referred to as single point extinction. This behavior is in sharp contrast with what happens when q ranges in $[p-1,p/2)$ and $p\in (2N/(N+1),2]$ for which we show complete extinction. Instantaneous shrinking and single point extinction take place in particular for the semilinear viscous Hamilton–Jacobi equation when $p=2$ and $q\in (0,1)$ and seem to have remained unnoticed.

Appendix: Proofs of Lemma 8.2 and gradient estimates for \(p=2\)

In this technical section we provide a fully rigorous proof of Lemma 8.2, as well as a gradient estimate for solutions to (1.1) for $p=2$. The latter, besides of its interest as an independent result, provides an essential technical tool in the proofs of our main results, and complements the gradient estimates in [25, Theorem 1.3], valid for $p\in (p_c,2)$.

Lemma 8.2 was proved in Sect. 8 at a formal level, presenting the essential calculations that give the ideas and essence of the proof, but allowing us at that point, for the simplicity of the exposition, to use results such as the maximum principle (or comparison principle) that are not automatically granted when we deal with singular coefficients. This is why we include the rigorous proof of this lemma here. To this end, we introduce a regularization of (1.1), already successfully used by the authors in [25, Sect. 6], in order to avoid the difficulties coming from the singularity at points where $\nabla u=0$. Let then u be a solution to the Cauchy problem (1.1)–(1.2) associated to an initial condition $u_0$ satisfying the assumptions (a)–(c) of Theorem 1.4. We recall that u vanishes identically after a finite time $T_e$ and that its positivity set $\mathcal {P}(t)$ is included in $B(0,R_0)$ for all $t\ge 0$, see (8.1).

For $\varepsilon \in (0,1/2)$, we define:

$$\begin{aligned} a_{\varepsilon }(z):=(z+\varepsilon ^2)^{(p-2)/2}, \quad b_{\varepsilon }(z):=(z+\varepsilon ^2)^{q/2}, \quad z\ge 0, \end{aligned}$$

(9.1)

and consider the following Cauchy problem:

$$\begin{aligned} \left\{ \begin{array}{ll}\partial _t \tilde{u}_{\varepsilon }-\mathrm{div}\left( a_{\varepsilon }(|\nabla \tilde{u}_{\varepsilon }|^2)\nabla \tilde{u}_{\varepsilon }\right) +b_{\varepsilon }\left( |\nabla \tilde{u}_{\varepsilon }|^2\right) -\varepsilon ^q =0, \quad (t,x)\in (0,\infty )\times \mathbb {R}^N, \\ \tilde{u}_{\varepsilon }(0)= u_{0, \varepsilon } +\varepsilon ^{\gamma },\quad x\in \mathbb {R}^N,\end{array}\right. \end{aligned}$$

(9.2)

where $\gamma \in (0,p/4)\cap (0,q/2)$ is a small positive parameter such that $\gamma <\min \{p-1,1-q\}$ and $u_{0,\varepsilon }\in C^{\infty }(\mathbb {R}^N)$ is a non-negative smooth approximation of the initial condition $u_0$, in the sense that $(u_{0,\varepsilon },\nabla u_{0,\varepsilon })$ converge to $(u_0,\nabla u_0)$ uniformly on compact sets in $\mathbb {R}^N$ and satisfy

$$\begin{aligned} 0 \le u_{0,\varepsilon } \le \Vert u_0\Vert _{\infty }, \quad \Vert \nabla u_{0,\varepsilon }\Vert _{\infty }\le (1+C(u_0)\varepsilon )\Vert \nabla u_0\Vert _{\infty }. \end{aligned}$$

(9.3)

It is proved in [5] and [25, Sect. 6] that (9.2) has a unique classical solution $\tilde{u}_\varepsilon$ and that, as $\varepsilon \rightarrow 0$, it is an approximation of the solution u to (1.1)–(1.2) with initial condition $u_0$ in the following sense:

$$\begin{aligned} u(t,x)=\lim \limits _{\varepsilon \rightarrow 0} \tilde{u}_{\varepsilon }(t,x), \quad \nabla u(t,x)=\lim \limits _{\varepsilon \rightarrow 0}\nabla \tilde{u}_{\varepsilon }(t,x), \end{aligned}$$

(9.4)

for almost every $(t,x)\in (0,\infty )\times \mathbb {R}^N$, the first convergence being actually uniform in compact sets of $(0,\infty )\times \mathbb {R}^N$. In addition, if $u_0$ is radially symmetric and non-increasing, then $u_{0,\varepsilon }$ can be chosen to be radially symmetric and non-increasing as well, so that $x\mapsto \tilde{u}_\varepsilon (t,x)$ is radially symmetric and non-increasing for any $t\ge 0$ and $\varepsilon \in (0,1/2)$. We next define

$$\begin{aligned} u_\varepsilon (t,x) := \tilde{u}_\varepsilon (t,x) - \varepsilon ^q t, \quad (t,x)\in (0,\infty )\times \mathbb {R}^N, \end{aligned}$$

and observe that the comparison principle and (9.2) imply that

$$\begin{aligned} u_\varepsilon (t,x) \ge \varepsilon ^\gamma - \varepsilon ^q t \ge \frac{\varepsilon ^\gamma }{2}, \quad (t,x)\in (0,\tau _\varepsilon )\times \mathbb {R}^N, \end{aligned}$$

(9.5)

with $\tau _\varepsilon := \varepsilon ^{\gamma -q}/2$ and that $u_\varepsilon$ solves

$$\begin{aligned} \left\{ \begin{array}{ll}\partial _t u_{\varepsilon }-\mathrm{div}\left( a_{\varepsilon }(|\nabla u_{\varepsilon }|^2)\nabla u_{\varepsilon }\right) +b_{\varepsilon }\left( |\nabla u_{\varepsilon }|^2\right) =0, \quad (t,x)\in (0,\infty )\times \mathbb {R}^N, \\ u_{\varepsilon }(0)= u_{0,\varepsilon } +\varepsilon ^{\gamma },\quad x\in \mathbb {R}^N.\end{array}\right. \end{aligned}$$

(9.6)

Since $\tau _\varepsilon \rightarrow \infty$ as $\varepsilon \rightarrow 0$, we may assume that $\varepsilon$ is taken sufficiently small to ensure $T_e\le \tau _\varepsilon$. With these approximations in mind, we are ready to give the complete proof of Lemma 8.2 as well as that of the gradient estimate (5.4) for $p=2$.

1.1 Proof of Lemma 8.2

Owing to the convergence properties of $u_{0,\varepsilon }$ and its gradient there holds

$$\begin{aligned} m_\varepsilon := \varepsilon + \Vert u_{0,\varepsilon } - u_0 \Vert _{C^1(B(0,R_0))} \mathop {\longrightarrow }_{\varepsilon \rightarrow 0} 0. \end{aligned}$$

(9.7)

Introducing

$$\begin{aligned} r_\varepsilon := \min \{ r \in [0,R_0) \, : \, u_0 (r) \le m_\varepsilon ^{1/4} \}, \end{aligned}$$

the properties of $u_0$ assumed in Theorem 1.4 and (9.7) ensure that there is $\varepsilon _0 \in (0, 1/2)$ such that

$$\begin{aligned} r_\varepsilon > s_\varepsilon := m_\varepsilon ^{(p-1-q)/4(p-q)} \;\; \text { and }\;\; m_\varepsilon \in (0,1) \;\;\text { for any }\;\; \varepsilon \in (0,\varepsilon _0). \end{aligned}$$

(9.8)

We then infer from the radial monotonicity of $u_0$, (1.12), (9.7), and (9.8) that, for $r\in [s_\varepsilon ,r_\varepsilon ]$,

$$\begin{aligned} |\partial _r u_{0,\varepsilon }(r)|&\ge |\partial _r u_0(r)| - m_\varepsilon \\&\ge \delta _0 r^{1/(p-1-q)} u_0(r)^{1/(p-q)} - m_\varepsilon \\&\ge \frac{\delta _0}{2} r^{1/(p-1-q)} u_{0,\varepsilon }^{1/(p-q)}(r) + \frac{\delta _0}{2} r^{1/(p-1-q)} \left( u_0^{1/(p-q)} - u_{0,\varepsilon }^{1/(p-q)} \right) (r) \\&\quad + \frac{\delta _0}{2} m_\varepsilon ^{1/2(p-q)} - m_\varepsilon \\&\ge \frac{\delta _0}{2} r^{1/(p-1-q)} u_{0,\varepsilon }^{1/(p-q)}(r) - \frac{\delta _0}{2} R_0^{1/(p-1-q)} m_\varepsilon ^{1/(p-q)} {+} \frac{\delta _0}{2} m_\varepsilon ^{1/2(p-q)} {-} m_\varepsilon \\&\ge \frac{\delta _0}{2} r^{1/(p-1-q)} u_{0,\varepsilon }^{1/(p-q)}(r) \\&\quad + m_\varepsilon ^{1/2(p-q)} \left( \frac{\delta _0}{2} - \frac{\delta _0}{2} R_0^{1/(p-1-q)} m_\varepsilon ^{1/2(p-q)} - m_\varepsilon ^{(2(p-q)-1)/ 2(p-q)} \right) . \end{aligned}$$

Since $p-q>1$, we infer from (9.7) and the above inequality that, taking $\varepsilon _0$ smaller if necessary, there holds

$$\begin{aligned} |\partial _r u_{0,\varepsilon } (r)| \ge \frac{\delta _0}{2} r^{1/(p-q-1)} u_{0,\varepsilon } (r)^{1/(p-q)}, \quad r \in [s_\varepsilon , r_\varepsilon ], \quad \varepsilon \in (0,\varepsilon _0). \end{aligned}$$

(9.9)

Now fix $\varepsilon \in (0, \varepsilon _0)$. Recalling that $a_\varepsilon$ and $b_\varepsilon$ are given by (9.1), we define

$$\begin{aligned} a_{1,\varepsilon }(z) := 2 z a_\varepsilon '(z) + a_\varepsilon (z), \quad z\ge 0, \end{aligned}$$

and the auxiliary function

$$\begin{aligned} J_\varepsilon (t,r):= & {} r^{N-1} a_\varepsilon (|\partial _r u_\varepsilon (t,r)|^2) \partial _r u_\varepsilon (t,r)\\&+ c(r) F(u_\varepsilon (t,r)), \quad (t,r) \in (0,T_e)\times (0,R_0). \end{aligned}$$

Since $u_\varepsilon$ solves (9.6) and

$$\begin{aligned} (p-1) a_\varepsilon (z) \le \ a_{1,\varepsilon }(z)&= (z+\varepsilon ^2)^{(p-4)/2} [(p-1)z+\varepsilon ^2] \le a_\varepsilon (z), \\ b_\varepsilon (z) - 2zb_\varepsilon '(z)&= (z+\varepsilon ^2)^{(q-2)/2} [(1-q) z + \varepsilon ^2] \ge (1-q) b_\varepsilon (z) \end{aligned}$$

for $z\ge 0$, we may perform the same computations as in the proof of Lemma 8.2 with $(a_\varepsilon ,b_\varepsilon )$ instead of (a, b) and derive the analogue of (8.7):

$$\begin{aligned} \partial _t J_\varepsilon - a_{1,\varepsilon }(g_\varepsilon ^2) \partial _r^2 J_\varepsilon + \left( \frac{N-1}{r} a_{1,\varepsilon }(g_\varepsilon ^2) - 2 b_\varepsilon '(g_\varepsilon ^2)g_\varepsilon \right) \partial _ r J_\varepsilon \le \sum _{i=1}^3 \mathcal {R}_{i,\varepsilon }, \end{aligned}$$

(9.10)

where $g_\varepsilon := - \partial _r u_\varepsilon \ge 0$ and

$$\begin{aligned} \mathcal {R}_{1,\varepsilon }&:= 2 \left[ (N-1) r^{N-2}a_\varepsilon (g_\varepsilon ^2) g_\varepsilon - c'(r) F(u_\varepsilon ) \right] b_\varepsilon '(g_\varepsilon ^2) g_\varepsilon , \\ \mathcal {R}_{2,\varepsilon }&:= \left[ \frac{N-1}{r} c'(r) - c''(r) \right] F(u_\varepsilon ) a_{1,\varepsilon }(g_\varepsilon ^2), \\ \mathcal {R}_{3,\varepsilon }&:= 2 \left[ c'(r) - \frac{N-1}{r} c(r) \right] F'(u_\varepsilon ) a_{1,\varepsilon }(g_\varepsilon ^2) g_\varepsilon - c(r) F''(u_\varepsilon ) a_{1,\varepsilon }(g_\varepsilon ^2) g_\varepsilon ^2 \\&\quad \;\; - (1-q) c(r) F'(u_\varepsilon ) b_\varepsilon (g_\varepsilon ^2). \end{aligned}$$

Observe that the positivity (9.5) of $u_\varepsilon$ guarantees that $F'(u_\varepsilon )$ and $F''(u_\varepsilon )$ are well-defined, even if F is not twice differentiable at zero. As in the proof of Lemma 8.2 we choose

$$\begin{aligned} c(r) = r^\lambda , \quad r\ge 0, \quad F(z) = \delta z^\beta , \quad z\ge 0, \end{aligned}$$

where $\delta >0$ is to be determined and

$$\begin{aligned} \lambda = N + \frac{q}{p-1-q} > N, \quad \beta = \frac{p-1}{p-q}\in (0,1). \end{aligned}$$

With this choice,

$$\begin{aligned} \mathcal {R}_{2,\varepsilon } \le - \lambda (\lambda -N) \delta (p-1) r^{\lambda -2} u_\varepsilon ^\beta a_{\varepsilon }(g_\varepsilon ^2), \end{aligned}$$

(9.11)

while

$$\begin{aligned} \mathcal {R}_{1,\varepsilon } = 2 r^{\lambda -1} b_\varepsilon '(g_\varepsilon ^2) g_\varepsilon \left[ (N-1) r^{N-1-\lambda } a_\varepsilon (g_\varepsilon ^2) g_\varepsilon - \delta \lambda u_\varepsilon ^\beta \right] \end{aligned}$$

(9.12)

and

$$\begin{aligned} \mathcal {R}_{3,\varepsilon }&\le \beta \delta \left[ \frac{2(\lambda -N+1)}{r} \left( g_\varepsilon ^2 + \varepsilon ^2 \right) ^{(p-1-q)/2} + \frac{1-\beta }{u_\varepsilon } \left( g_\varepsilon ^2 + \varepsilon ^2 \right) ^{(p-q)/2} \right. \nonumber \\&\quad - (1-q) \Big ] r^\lambda u_\varepsilon ^{\beta -1} b_\varepsilon (g_\varepsilon ^2). \end{aligned}$$

(9.13)

Now let $\kappa > 0$ and define

$$\begin{aligned} \mathcal {J}_{\kappa ,\varepsilon } := \{ (t,r)\in (0,T_e)\times (0,R_0)\ :\ J_\varepsilon (t,r) \ge \kappa \}. \end{aligned}$$

Then

$$\begin{aligned} \kappa + r^{N-1} a_\varepsilon (g_\varepsilon ^2) g_\varepsilon \le c(r) F(u_\varepsilon ) = \delta r^\lambda u_\varepsilon ^\beta \;\;\text { in }\;\; \mathcal {J}_{\kappa ,\varepsilon }, \end{aligned}$$

from which we deduce that, if $\kappa \ge R_0^{N-1} \varepsilon ^{p-1}$, then

$$\begin{aligned} r^{N-1} (g_\varepsilon ^2 + \varepsilon ^2 )^{(p-1)/2} \le r^{N-1} a_\varepsilon (g_\varepsilon ^2) g_\varepsilon + R_0^{N-1} \varepsilon ^{p-1} \le \delta r^\lambda u_\varepsilon ^\beta \;\;\text { in }\;\; \mathcal {J}_{\kappa ,\varepsilon }. \end{aligned}$$

(9.14)

This inequality implies in particular that $r>0$ and $u_\varepsilon >0$ in $\mathcal {J}_{\kappa ,\varepsilon }$ and (9.11) yields

$$\begin{aligned} {\mathcal {R}_{2,\varepsilon }} \le - \lambda (\lambda -N)\delta (p-1) r^{\lambda -2} u_\varepsilon ^\beta \left( g_\varepsilon ^2 + \varepsilon ^2 \right) ^{(p-2)/2} <0 \;\;\text { in }\;\; \mathcal {J}_{\kappa ,\varepsilon } \end{aligned}$$

(9.15)

in view of $\lambda >N$ and $p \in (1,2]$. We then infer from (9.12), (9.13), and (9.14) that, in $\mathcal {J}_{\kappa ,\varepsilon }$,

$$\begin{aligned} \mathcal {R}_{1,\varepsilon } \le - 2 (\lambda -N+1) \delta r^{\lambda -1} u_\varepsilon ^\beta b_\varepsilon '(g_\varepsilon ^2) g_\varepsilon \le 0 \end{aligned}$$

(9.16)

and

$$\begin{aligned} \mathcal {R}_{3,\varepsilon }&\le \beta \delta \left[ 2(\lambda -N+1) \delta ^{(p-1-q)/(p-1)} \left( \Vert u_0\Vert _\infty +\varepsilon ^\gamma \right) ^{(p-1-q)/(p-q)} \right. \nonumber \\&\quad \left. + (1-\beta ) \delta ^{(p-q)/(p-1)} R_0^{(p-q)/(p-1-q)} - (1-q) \right] r^\lambda u_\varepsilon ^{\beta -1} b_\varepsilon (g_\varepsilon ^2) \nonumber \\&\le - \frac{\beta \delta (1-q)}{2} r^\lambda u_\varepsilon ^{\beta -1} b_\varepsilon (g_\varepsilon ^2) \le 0, \end{aligned}$$

(9.17)

provided $\delta$ is chosen suitably small (depending on $\Vert u_0\Vert _\infty$ and $R_0$) and independent of $\varepsilon \in (0,\varepsilon _0)$ as $\Vert u_0\Vert _\infty +\varepsilon ^\gamma \le \Vert u_0\Vert _\infty +1$. Collecting (9.10), (9.15), (9.16), and (9.17), we end up with

$$\begin{aligned} \partial _t J_\varepsilon - a_{1,\varepsilon }(g_\varepsilon ^2) \partial _r^2 J_\varepsilon + \left( \frac{N-1}{r} a_{1,\varepsilon }(g_\varepsilon ^2) - 2 b_\varepsilon '(g_\varepsilon ^2)g_\varepsilon \right) \partial _ r J_\varepsilon < 0, \end{aligned}$$

(9.18)

for $(t,r)\in \mathcal {J}_{\kappa ,\varepsilon }$, this inequality being true only for $\kappa \ge R_0^{N-1} \varepsilon ^{p-1}$.

Next, introducing

$$\begin{aligned} M_\varepsilon := \sup _{t\in [0,T_e]} u_\varepsilon (t,R_0), \end{aligned}$$

we infer from the monotonicity of $r\mapsto u_\varepsilon (t,r)$ that

$$\begin{aligned} J_\varepsilon (t,0)=0, \quad J_\varepsilon (t,R_0) \le \delta R_0^\lambda M_\varepsilon ^\beta , \quad t\in [0,T_e]. \end{aligned}$$

(9.19)

In addition, given $r\in (0,R_0)$,

$$\begin{aligned} J_\varepsilon (0,r) \le r^{N-1} a_\varepsilon (|\partial _r u_{0,\varepsilon }(r)|^2) \partial _r u_{0,\varepsilon }(r) + \delta r^\lambda u_{0,\varepsilon }^\beta (r) + \delta R_0^\lambda \varepsilon ^{\gamma \beta }. \end{aligned}$$

As $\partial _r u_{0,\varepsilon }(r) \le 0$, we obtain from (9.3) and (9.7) that

$$\begin{aligned} J_\varepsilon (0,r) \le \delta s_\varepsilon ^\lambda \Vert u_0 \Vert _\infty ^\beta + \delta R_0^\lambda \varepsilon ^{\gamma \beta }, \quad r \in (0,s_\varepsilon ), \end{aligned}$$

(9.20)

as well as

$$\begin{aligned} J_\varepsilon (0,r) \le \delta R_0^\lambda \left( m_\varepsilon + m_\varepsilon ^{1/4} \right) ^\beta + \delta R_0^\lambda \varepsilon ^{\gamma \beta }, \quad r \in (r_\varepsilon , R_0). \end{aligned}$$

(9.21)

For $r \in [s_\varepsilon , r_\varepsilon ]$, we now divide the analysis into two regions with respect to the magnitude of $|\partial _r u_{0,\varepsilon }(r)|$. Either $|\partial _r u_{0,\varepsilon }(r)| \le \varepsilon$ and we deduce from (9.9) that

$$\begin{aligned} u_{0,\varepsilon }(r)^{1/(p-q)}\le \frac{2\varepsilon }{\delta _0}r^{-1/(p-q-1)}. \end{aligned}$$

Thus, taking also into account that $\partial _r u_{0,\varepsilon }(r)\le 0$ for any $r\ge 0$, we realize that

$$\begin{aligned} \begin{array}{ll} J_\varepsilon (0,r) &{} \le \delta r^\lambda (2 \varepsilon )^{p-1} \delta _0^{1-p} r^{-(p-1)/(p-1-q)} + \delta R_0^\lambda \varepsilon ^{\gamma \beta } \\ &{} \le \delta \delta _0^{1-p} R_0^{N-1} (2 \varepsilon )^{p-1} + \delta R_0^\lambda \varepsilon ^{\gamma \beta }. \end{array} \end{aligned}$$

(9.22)

Or $|\partial _r u_{0,\varepsilon }(r)| > \varepsilon$ which implies that

$$\begin{aligned} a_\varepsilon (|\partial _r u_{0,\varepsilon }(r)|^2) \ge 2^{(p-2)/2} |\partial _r u_{0,\varepsilon }(r)|^{p-2}. \end{aligned}$$

Therefore, using again (9.9),

$$\begin{aligned} \begin{array}{ll} J_\varepsilon (0,r) &{} \le \delta r^\lambda u_{0,\varepsilon }(r)^\beta - 2^{(p-2)/2} r^{N-1} |\partial _r u_{0,\varepsilon }(r)|^{p-1} + \delta R_0^\lambda \varepsilon ^{\gamma \beta } \\ &{} \le \left( \delta - 2^{-p/2} \delta _0^{p-1} \right) r^\lambda u_{0,\varepsilon }(r)^\beta + \delta R_0^\lambda \varepsilon ^{\gamma \beta } \le \delta R_0^\lambda \varepsilon ^{\gamma \beta }, \end{array} \end{aligned}$$

(9.23)

provided $\delta < 2^{-p/2} \delta _0^{p-1}$. In view of (9.8) and (9.20)–(9.23) we have thus shown that, if $\delta$ is sufficiently small,

$$\begin{aligned} J_\varepsilon (0,r) \le \delta \delta _0^{1-p} R_0^{N-1} (2\varepsilon )^{p-1} + \delta \Vert u_0 \Vert _\infty ^\beta s_\varepsilon ^\lambda + \delta R_0^\lambda \left( {2^\beta m_\varepsilon ^{\beta /4}} + \varepsilon ^{\gamma \beta } \right) , \quad r\in (0,R_0). \end{aligned}$$

Consequently, if $\delta$ is sufficiently small and

$$\begin{aligned} \kappa&= \kappa _\varepsilon \\&:= \delta \delta _0^{1-p} R_0^{N-1} (2\varepsilon )^{p-1} + \delta \Vert u_0 \Vert _\infty ^\beta s_\varepsilon ^\lambda + \delta R_0^\lambda \left( {2^\beta m_\varepsilon ^{\beta /4}} + \varepsilon ^{\gamma \beta } \right) \\&\quad + R_0^{N-1} \varepsilon ^{p-1} + \delta R_0^\lambda M_\varepsilon ^\beta , \end{aligned}$$

then the parabolic boundary $\{0\}\times (0,R_0)$ and $[0,T_e) \times \{0,R_0\}$ of $(0,T_e)\times (0,R_0)$ contains no point in $\mathcal {J}_{\kappa _\varepsilon ,\varepsilon }$. Recalling (9.18) we may then argue as in the proof of Lemma 8.2 to conclude that

$$\begin{aligned} J_\varepsilon \le \kappa _\varepsilon \;\;\text { in }\;\; (0,T_e)\times (0,R_0). \end{aligned}$$

(9.24)

To complete the proof, we observe that $M_\varepsilon \rightarrow 0$ as $\varepsilon \rightarrow 0$ due to the uniform convergence (9.4) and the vanishing of u on $(0,T_e)\times \partial B(0,R_0)$. Combining this fact with (9.7) and (9.8) yields

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \kappa _\varepsilon =0, \end{aligned}$$

and we may let $\varepsilon \rightarrow 0$ in (9.24) and use (9.4) and (9.7) to obtain the expected result.

1.2 Proof of (5.4) for $p=2$

Finally, we prove the gradient estimate (5.4) for $p=2$.

Proposition 9.1

Consider an initial condition $u_0$ satisfying (1.3) and denote the corresponding solution to (1.1)–(1.2) by u. Assume further that $p=2$ and $q \in (0,1)$. Then there is $C_1>0$ depending only on q such that

$$\begin{aligned} \left| \nabla u^{(1-q)/(2-q)}(t,x)\right| \le C_1\left( 1+\Vert u_0\Vert _{\infty }^{(1-q)/(2-q)}t^{-1/2}\right) , \end{aligned}$$

(9.25)

for $(t,x)\in (0,\infty )\times \mathbb {R}^N$.

Proof

We fix $\varepsilon \in (0, 1/2)$ and denote the classical solution to (9.2) by $\tilde{u}_\varepsilon$. Observe that $a_\varepsilon \equiv 1$ due to $p=2$. In view of (9.3), the comparison principle implies

$$\begin{aligned} \varepsilon ^\gamma \le \tilde{u}_\varepsilon (t,x) \le \Vert u_0 \Vert _\infty + \varepsilon ^\gamma , \quad (t,x) \in [0,\infty ) \times \mathbb {R}^N. \end{aligned}$$

(9.26)

We further set

$$\begin{aligned} f(\xi ) := \frac{1-q}{2-q} \xi ^{(2-q)/(1-q)}, \; \xi \ge 0, \quad v_\varepsilon := f^{-1} (\tilde{u}_\varepsilon ), \quad w_\varepsilon := |\nabla v_\varepsilon |^2, \end{aligned}$$

and note that $f \in C^2([0,\infty )) \cap C^\infty ((0,\infty ))$ is strictly increasing. Hence, according to [5, Formula (10)], we have

$$\begin{aligned} \mathcal{P}_\varepsilon w_\varepsilon \le 2 \left( \frac{f^{\prime \prime }}{f^\prime } \right) ^\prime (v_\varepsilon ) w_\varepsilon ^2 - 2 \left( \frac{f^{\prime \prime }}{(f^\prime )^2} \right) (v_\varepsilon ) \Theta _\varepsilon \left( (f^\prime )^2 (v_\varepsilon ) w_\varepsilon \right) w_\varepsilon \end{aligned}$$

(9.27)

in $(0,\infty ) \times \mathbb {R}^N$, where

$$\begin{aligned} \mathcal{P}_\varepsilon w_\varepsilon&:= \partial _t w_\varepsilon - \Delta w_\varepsilon + 2 \left( f^\prime (v_\varepsilon ) b_\varepsilon ^\prime \left( (f^\prime )^2 (v_\varepsilon ) w_\varepsilon \right) - \left( \frac{f^{\prime \prime }}{f^\prime } \right) (v_\varepsilon ) \right) \nabla v_\varepsilon \cdot \nabla w_\varepsilon ,\\ \Theta _\varepsilon (\xi )&:= 2\xi b_\varepsilon ^\prime (\xi ) - b_\varepsilon (\xi ) + \varepsilon ^q, \quad \xi \ge 0. \end{aligned}$$

Since $q \in (0,1)$, we have

$$\begin{aligned} \Theta _\varepsilon (\xi )&= q \xi (\xi + \varepsilon ^2)^{(q-2)/2} - (\xi + \varepsilon ^2)^{q/2} + \varepsilon ^q \\&= -(1-q) (\xi + \varepsilon ^2)^{q/2} - q \varepsilon ^2 (\xi + \varepsilon ^2)^{(q-2)/2} + \varepsilon ^q \\&\ge -(1-q) (\xi + \varepsilon ^2)^{q/2} + (1-q) \varepsilon ^q \\&\ge -(1-q) \xi ^{q/2}, \quad \xi \ge 0. \end{aligned}$$

Hence, (9.27), the choice of f, the nonnegativity of $w_\varepsilon$, and Young’s inequality imply

$$\begin{aligned} \mathcal{P}_\varepsilon w_\varepsilon&\le - \frac{2}{(1-q) v_\varepsilon ^2} w_\varepsilon ^2 + 2 \frac{1}{1-q} v_\varepsilon ^{(q-2)/(1-q)} (1-q) \left( v_\varepsilon ^{2/(1-q)} w_\varepsilon \right) ^{q/2} w_\varepsilon \nonumber \\&\le - \frac{2}{(1-q) v_\varepsilon ^2} w_\varepsilon ^2 + \frac{2}{(1-q)v_\varepsilon ^2} w_\varepsilon ^{1+ q/2}\nonumber \\&\le \frac{2}{(1-q) v_\varepsilon ^2} \left( - w_\varepsilon ^2 + \frac{2+q}{4} w_\varepsilon ^2 + \frac{2-q}{4} \right) \nonumber \\&= - \frac{2-q}{2(1-q) v_\varepsilon ^2} \left( w_\varepsilon ^2 -1 \right) . \end{aligned}$$

(9.28)

Considering next

$$\begin{aligned} W(t) := 1 + \frac{a}{t}, \; t >0, \quad \text{ with } \; a := 2 \left( \frac{1-q}{2-q} \right) ^{q/(2-q)} (\Vert u_0\Vert _\infty +1)^{2(1-q)/(2-q)}, \end{aligned}$$

noticing that (9.26) implies

$$\begin{aligned} v_\varepsilon = \left( \frac{2-q}{1-q} \tilde{u}_\varepsilon \right) ^{(1-q)/(2-q)} \le \left( \frac{2-q}{1-q} (\Vert u_0\Vert _\infty +1) \right) ^{(1-q)/(2-q)}, \end{aligned}$$

and using the fact that $a>0$, we obtain

$$\begin{aligned}&\mathcal{P}_\varepsilon W + \frac{2-q}{2(1-q) v_\varepsilon ^2} \left( W^2 -1 \right) \\&\quad \ge -\frac{a}{t^2} + \frac{2-q}{2(1-q) v_\varepsilon ^2} \cdot \frac{a^2}{t^2} \\&\quad \ge \frac{a}{t^2} \left[ -1 + \frac{a}{2} \left( \frac{2-q}{1-q} \right) ^{q/(2-q)} (\Vert u_0\Vert _\infty +1)^{-2(1-q)/(2-q)} \right] \ge 0 \end{aligned}$$

for any $t>0$. Since $w_\varepsilon (0,x) < W(0) = \infty$ for $x \in \mathbb {R}^N$, we deduce from (9.28) and the comparison principle that

$$\begin{aligned} \left( \frac{2-q}{1-q} \right) ^{(1-q)/(2-q)} \left| \nabla \tilde{u}_\varepsilon ^{(1-q)/(2-q)} (t,x) \right| = |\nabla v_\varepsilon (t,x) | = w_\varepsilon ^{1/2} (t,x) \le \left( 1+ \frac{a}{t} \right) ^{1/2} \end{aligned}$$

in $(0,\infty ) \times \mathbb {R}^N$. Letting $\varepsilon \searrow 0$ and recalling (9.4), we end up with (9.25).$\square$