In this section, we examine the role of (1.2). First of all, we present a standard lemma which asserts that (1.2) alone is sufficient to propagate positivity of u in measure for a short period of time (cf. [13]).
Proposition 4.1
Suppose u is nonnegative and satisfies (1.2)(_{-}). Assume for (M>0) and (alpha in (0,1)), we have ((s,s+varrho ^{p}]times K_{varrho }(y)subset E) and
$$begin{aligned} |[u(cdot , s)>M]cap K_{varrho }(y)|ge alpha |K_{varrho }|. end{aligned}$$
Then, there exist (delta , varepsilon in (0,1)) depending only on the data and (alpha ), such that
$$begin{aligned} |[u(cdot , t)>varepsilon M]cap K_{varrho }(y)|ge tfrac{1}{2}alpha |K_{varrho }| end{aligned}$$
for all times
$$begin{aligned} s<t<s+delta varrho ^{p}. end{aligned}$$
Proof
Assume ((y,s)=(0,0)). We may apply (1.2)(_{-}) with (k=M) in the cylinders
in such a case, we have for all (0<t<delta varrho ^{p}),
$$begin{aligned} int _{K_{(1-sigma )varrho }} (u(cdot ,t)-M)^{p}_- mathrm {d}x&le int _{K_{varrho }} (u(x,0)-M)^{p}_- mathrm {d}x+frac{gamma }{(sigma varrho )^{p}}iint _{Q_{o}}(u-M)^{p}_- mathrm {d}xmathrm {d}t\ {}&le int _{K_{varrho }} (u(x,0)-M)^{p}_- mathrm {d}x+gamma frac{M^{p}}{(sigma varrho )^{p}} |[u<M]cap Q_{o}|\ {}&le M^{p}left[ 1-alpha +gamma frac{delta }{sigma ^{p}}frac{|[u<M]cap Q_{o}|}{|Q_{o}|}right] |K_{varrho }|. end{aligned}$$
Set (ell =varepsilon M). The left-hand side of the above estimate can be bounded from below by
$$begin{aligned} int _{K_{(1-sigma )varrho }cap [ule ell ]} (u(cdot ,t)-M)^{p}_- mathrm {d}xge (1-varepsilon )^{p} M^{p}|A_{ell ,(1-sigma )varrho }(t)| end{aligned}$$
where we have defined, for some (varepsilon ) to be chosen, that
$$begin{aligned} A_{ell ,(1-sigma )varrho }(t)=[u(cdot ,t)le varepsilon M]cap K_{(1-sigma )varrho }. end{aligned}$$
Notice that
$$begin{aligned} |A_{ell ,varrho }(t)|&=|A_{ell ,(1-sigma )varrho }(t)cup (A_{ell ,varrho }(t) -A_{ell ,(1-sigma )varrho }(t))|\&le |A_{ell ,(1-sigma )varrho }(t)|+|K_{varrho }- K_{(1-sigma )varrho }|\&le |A_{ell ,(1-sigma )varrho }(t)|+Nsigma |K_{varrho }|. end{aligned}$$
Collecting all the above estimates yields that
$$begin{aligned} |A_{ell ,varrho }(t)|le frac{1-alpha }{(1-varepsilon )^{p}}|K_{varrho }| +Cfrac{delta }{sigma ^{p}}frac{|[u<M]cap Q_{o}|}{|Q_{o}|}|K_{varrho }| +Nsigma |K_{varrho }| end{aligned}$$
(4.1)
Finally, we may choose (varepsilon ), (sigma ) and (delta ), such that
$$begin{aligned} frac{1-alpha }{(1-varepsilon )^{p}}le 1-tfrac{3}{4}alpha ,quad Nsigma =tfrac{1}{8}alpha ,quad Cfrac{delta }{sigma ^{p}}le tfrac{1}{8}alpha . end{aligned}$$
(square )
Remark 4.1
One easily obtains the dependence of various constants on (alpha ) from the above proof, namely (varepsilon approx alpha ), (sigma approx alpha ) and (delta approx alpha ^{p+1}).
One wonders if the positivity in measure can be propagated further in time, i.e., (delta ) can be made large by choosing a proper (varepsilon ). It seems (1.2)(_{-}) alone is insufficient. In the theory of parabolic equations, a standard tool to achieve this is a logarithmic estimate. See [3], Chapter 2, Section 3]. We do not know if such a logarithmic estimate holds for functions in parabolic De Giorgi classes. However, we show in the following that a membership in (uin {mathfrak {B}}^{-}_{p}(E,gamma )) still ensures that the measure information of positivity propagates further in time.
Proposition 4.2
Suppose (uin {mathfrak {B}}^{-}_{p}(E,gamma )) is nonnegative. Assume for (A, M>0) and (alpha in (0,1)), we have ((s,s+Avarrho ^{p}]times K_{varrho }(y)subset E) and
$$begin{aligned} |[u(cdot , s)>M]cap K_{varrho }(y)|ge alpha |K_{varrho }|. end{aligned}$$
Then, there exist (varepsilon >0) depending on the data, A and (alpha ), such that
$$begin{aligned} |[u(cdot , t)>varepsilon M]cap K_{varrho }(y)|ge tfrac{1}{2}alpha |K_{varrho }| end{aligned}$$
for all
$$begin{aligned} s<t<s+Avarrho ^{p}. end{aligned}$$
Shrinking the measure of the set ([uapprox 0])
We first prove the following shrinking lemma due to De Giorgi (cf. [1]).
Lemma 4.1
Let (alpha , delta in (0,1)). Suppose there holds
$$begin{aligned} left| left[ u(cdot , t)>Mright] cap K_{varrho }right| ge alpha |K_{varrho }| quad text { for all }tin (s,s+delta varrho ^{p}]. end{aligned}$$
There exists (C>0) depending only on the data, such that for any positive integer (j_{*}), we have
$$begin{aligned} left| left[ ule frac{M}{2^{j_{*}}}right] cap Qright| le frac{C}{alpha delta ^{frac{1}{p}} j_{*}^{frac{p-1}{p}}}|Q|,quad text {where}Q=K_{varrho }times left( s,s+delta varrho ^{p}right] . end{aligned}$$
Proof
We assume ((y,s)=(0,0)) and set (k_j=2^{-j}M) for (j=0,1,ldots , j_{*}). Apply (1.1)(_{-}) for the pair of cylinders
$$begin{aligned} K_{varrho }times (0,delta varrho ^{p}]subset K_{2varrho }times (-delta varrho ^{p},delta varrho ^{p}], end{aligned}$$
such that
$$begin{aligned} iint _{Q}|D(u-k_j)_-|^{p} mathrm {d}xmathrm {d}tle frac{C}{delta varrho ^{p}}left( frac{M}{2^j}right) ^{p}|Q|. end{aligned}$$
(4.2)
Next, we apply [3], Chapter I, Lemma 2.2] to (u(cdot ,t)) for (tin left( 0,delta varrho ^{p}right] ) over the cube (K_{varrho }), for levels (k_{j+1}<k_{j}). Taking into account the measure theoretical information
$$begin{aligned} left| left[ u(cdot , t)>Mright] cap K_{varrho }right| ge alpha |K_{varrho }| quad text{ for } text{ all } tin (0,delta varrho ^{p}], end{aligned}$$
this gives
$$begin{aligned} frac{M}{2^{j+1}}&|[u(cdot ,t)<k_{j+1}]cap K_{varrho }|\&le frac{C varrho ^{N+1}}{|[u(cdot ,t)>k_j]cap K_{varrho }|}int _{[k_j<u(cdot ,t)<k_{j+1}] cap K_{varrho }}|Du| mathrm {d}x\&le frac{Cvarrho }{alpha }bigg (int _{[k_j<u(cdot ,t)<k_{j+1}]cap K_{varrho }}|Du|^{p} mathrm {d}xbigg )^{frac{1}{p}}\&quad times |([u(cdot ,t)<k_j]-[u(cdot ,t)<k_{j+1}])cap K_{varrho }|^{frac{p-1}{p}}. end{aligned}$$
Set
$$begin{aligned} A_j=[u<k_j]cap Q end{aligned}$$
and integrate the above estimate in (mathrm {d}t) over ((0,delta varrho ^{p}]); we obtain by using (4.2)
$$begin{aligned} frac{M}{2^j}|A_{j+1}|&le frac{Cvarrho }{alpha }bigg (iint _{Q}|D(u-k_j)_-|^{p} mathrm {d}xmathrm {d}tbigg )^frac{1}{p}(|A_j|-|A_{j+1}|)^frac{p-1}{p}\&le frac{C}{alpha delta ^{frac{1}{p}}}frac{M}{2^j}|Q|^{frac{1}{p}}(|A_j|-|A_{j+1}|)^frac{p-1}{p}. end{aligned}$$
Now take the power (frac{p}{p-1}) on both sides of the above inequality to obtain
$$begin{aligned} |A_{j+1}|^{frac{p}{p-1}}le frac{C}{alpha ^{frac{p}{p-1}} delta ^{frac{1}{p-1}}}|Q|^{frac{1}{p-1}}(|A_j|-|A_{j+1}|). end{aligned}$$
Add these inequalities from 0 to (j_{*}-1) to obtain
$$begin{aligned} j_{*} |A_{j_{*}}|^{frac{p}{p-1}}le sum _{j=0}^{j_{*}-1}|A_{j+1}|^{frac{p}{p-1}} le frac{C}{alpha ^{frac{p}{p-1}}delta ^{frac{1}{p-1}}}|Q|^{frac{p}{p-1}}. end{aligned}$$
From this, we conclude
$$begin{aligned} |A_{j_{*}}|le frac{C}{alpha delta ^{frac{1}{p}} j_{*}^{frac{p-1}{p}}}|Q|. end{aligned}$$
(square )
Proof of Proposition 4.2
We come back at (4.1) and choose
$$begin{aligned} sigma =delta ^{frac{1}{p+1}}left( frac{|[u<k]cap Q_{o}|}{|Q_{o}|}right) ^{frac{1}{p+1}}, end{aligned}$$
such that (4.1) becomes
$$begin{aligned} |A_{ell ,varrho }(t)|le bigg [frac{1-alpha }{(1-varepsilon )^{p}} +Cdelta ^{frac{1}{p+1}}left( frac{|[u<k]cap Q_{o}|}{|Q_{o}|}right) ^{frac{1}{p+1}} bigg ]|K_{varrho }|. end{aligned}$$
We choose (delta ) and (varepsilon ) such that
$$begin{aligned} Cdelta ^{frac{1}{p+1}}=tfrac{1}{8}alpha , quad frac{1-alpha }{(1-varepsilon )^{p}}<frac{1-frac{1}{2}alpha }{(1-varepsilon )^{p}}le 1-tfrac{1}{4}alpha . end{aligned}$$
(4.3)
As a result, we obtain
Having (varepsilon ) and (delta ) determined in (4.3), we use (1.2)(_{-}) again and repeat the above argument with
$$begin{aligned} M_{1}=varepsilon M,quad ell _{1}=frac{M_{1}}{2^{n_{1}+j_{1}}},quad k_{1}=frac{M_{1}}{2^{j_{1}}}, end{aligned}$$
where (j_{1}) and (n_{1}) are positive numbers to be determined. We may use the above measure theoretical information for (tin [s,s_{1}]), and apply Lemma 4.1 to obtain a refined estimate:
$$begin{aligned} |A_{ell _{1},varrho }(t)| le Bigg [frac{1-alpha }{(1- 2^{-n_{1}})^{2}} +Cdelta ^{frac{1}{p+1}}left( frac{1}{alpha delta ^{frac{1}{p}}j_{1}^{frac{p-1}{p}}}right) ^{frac{1}{p+1}}Bigg ]|K_{varrho }| quad text { for all },, tin [0, s_{1}]. end{aligned}$$
We choose (j_{1}) and (n_{1}), such that
$$begin{aligned} Cdelta ^{frac{1}{p+1}}left( frac{1}{alpha delta ^{frac{1}{p}}j_{1}^{frac{p-1}{p}}}right) ^{frac{1}{p+1}}le frac{delta alpha }{4A}, quad frac{1-alpha }{(1-2^{-n_{1}})^{2}}le 1-alpha +frac{delta alpha }{4A}. end{aligned}$$
As a result, we obtain that
$$begin{aligned} |A_{ell _{1},varrho }(t)|le left( 1-alpha +frac{delta alpha }{2A}right) |K_{varrho }| quad text { for all }tin [s,s_{1}]. end{aligned}$$
Now, we may proceed by induction. Suppose the construction has been made up to the ((i-1))-th step: the sequences ({M_{i}}), ({n_{i}}) and ({j_{i}}) have been chosen up to the ((i-1))-th step, and we have the measure theoretical information
$$begin{aligned} |A_{ell _{i-1},varrho }(t)|le left( 1-alpha +(i-1)frac{delta alpha }{2A}right) |K_{varrho }| quad text { for all }tin [s_{i-1},s_{i}], end{aligned}$$
where
Setting
$$begin{aligned} ell ^{varepsilon }_{i-1}=varepsilon widehat{M}_{i-1},quad s_{i+1}=s_{i}+delta varrho ^{p},quad Q_{i}=K_{varrho }times (s_{i}, s_{i+1}], end{aligned}$$
and using the above measure theoretical information at (t=s_{i}), we can repeat the above argument to obtain that, for all (tin [s_{i},s_{i+1}]),
$$begin{aligned} |A_{ell ^{varepsilon }_{i-1},varrho }(t)|le bigg [frac{1-alpha +(i-1)frac{1}{2A}delta alpha }{(1-varepsilon )^{2}} +Cdelta ^{frac{1}{p+1}}left( frac{|[u<ell _{i-1}]cap Q_{i}|}{|Q_{i}|}right) ^{frac{1}{p+1}}bigg ]|K_{varrho }|. end{aligned}$$
Assuming ((i-1)delta <A), we may choose (varepsilon ) and (delta ) as in (4.3); this ensures
$$begin{aligned} |A_{ell ^{varepsilon }_{i-1},varrho }(t)|le left( 1-tfrac{1}{8}alpha right) |K_{varrho } |quad text { for all }tin [s_{i},s_{i+1}]. end{aligned}$$
Now, we set
$$begin{aligned}&M_{i}=varepsilon widehat{M}_{i-1},quad ell _{i}=frac{ M_{i}}{2^{n_{i}+j_{i}}},quad k_{i}=frac{M_{i}}{2^{j_{i}}}, end{aligned}$$
where (j_{i}) and (n_{i}) are to be determined. Then, we use the above measure theoretical information in Lemma 4.1 to obtain a refined estimate: for all (tin [s_{i},s_{i+1}]),
$$begin{aligned} |A_{ell _{i},varrho }(t)|&le left[ frac{1-alpha +(i-1)frac{1}{2A}delta alpha }{(1-2^{-n_{i}})^{2}} +Cdelta ^{frac{1}{p+1}}left( frac{1}{alpha delta ^{frac{1}{p}}j_{i}^{frac{p-1}{p}}}right) ^ {frac{1}{p+1}}right] |K_{varrho }|. end{aligned}$$
We choose (j_{i}) and (n_{i}), such that
$$begin{aligned}&Cdelta ^{frac{1}{p+1}}left( frac{1}{alpha delta ^{frac{1}{p}}j_{i}^{frac{p-1}{p}}}right) ^{frac{1}{p+1}}le frac{delta alpha }{4A},\&frac{1-alpha +(i-1)frac{1}{2A}delta alpha }{(1-2^{-n_{i}})^{2}}le 1-alpha +(i-1)frac{delta alpha }{2A}+frac{delta alpha }{4A}. end{aligned}$$
As a result, we obtain that for all times (tin [s_{i},s_{i+1}]),
$$begin{aligned} |A_{ell _{i},varrho }(t)|le left( 1-alpha +ifrac{delta alpha }{2A}right) |K_{varrho }|. end{aligned}$$
The above argument terminates if (idelta ge A), and we reach the desired conclusion with the choice
$$begin{aligned} varepsilon M=frac{ M_{i}}{2^{n_{i}+j_{i}}}. end{aligned}$$
