Abstract
We study local boundedness and Hölder continuity of a parabolic equation involving the fractional p-Laplacian of order s, with \(0<s<1\), \(2\le p < \infty \), with a general right-hand side. We focus on obtaining precise Hölder continuity estimates. The proof is based on a perturbative argument using the already known Hölder continuity estimate for solutions to the equation with zero right-hand side.
Similar content being viewed by others
1 Introduction
In this paper, we study the local boundedness and Hölder regularity of solutions to the inhomogeneous equation
where \(f \in L^r_{\textrm{loc}}(I; L^q_{\textrm{loc}}(\Omega ))\) with \(q\ge 1\), \(r\ge 1\), \(p\ge 2\) and \(s \in (0,1)\). Here, \( (- \Delta _p)^s \) is the fractional p-Laplacian, arising as the first variation of the Sobolev–Slobodeckiĭ seminorm
Nonlocal equations involving operators of the above type, with a singular kernel, were first considered in [31] to the best of our knowledge.
In this study, continuing the work in [7], we perform a perturbative argument to obtain Hölder continuity estimates, with explicit exponents for the equations with a right-hand side. Our approach closely follows the arguments in [47] and [6]. In such perturbative arguments, it is often possible to establish Hölder regularity results for bounded solutions using only \(L^\infty \) estimates for the equations with zero right-hand side. Here, to estimate the Hölder seminorms of certain functions in the proof of Theorem 1.2 as well as to prove Theorem 3.6, we are led to prove Proposition 3.4. As a by-product, by combining Proposition 3.4 with the existing local boundedness results we obtain an \(L^\infty \) bound for equations with right-hand sides. This is Theorem 1.1. The proof is inspired by the work [5].
Below, we state the main results. For the definition of the tail and relevant function spaces, see Sect. 2. We use the following notation of parabolic cylinders
The exponent \(p_s^\star = \frac{np}{n-sp}\) is the critical exponent for the Sobolev embedding theorem, see Proposition 2.5. We denote by \(p^\prime \), the Hölder conjugate of p, that is \(p^\prime = \frac{p}{p-1}\).
Theorem 1.1
Let \(\Omega \subset {\mathbb {R}}^n\) be a bounded and open set, \(I=(t_0,t_1]\), \(p\ge 2\), \(0<s<1\). Consider q and r such that
In addition, assume that \(r\ge p^\prime \),
Suppose u is a local weak solution of
such that
then u is locally bounded in \(\Omega \). More specifically, if \(Q_{2R,(2R)^{sp}(x_0,T_0)} \Subset \Omega \times I\), u is bounded in \(Q_{R/2,(R/2)^{sp}}(x_0,T_0)\), and in the case \(sp \ne n \), the estimate reads
where \(C=C(n,s,p)\), \(\nu = 1-\frac{1}{r} - \frac{n}{spq}\) and \( \vartheta = 1+\frac{sp\nu }{n} \).
In the case \(sp=n \), given any l such that \( \frac{p}{r^\prime }(1-\frac{1}{r} - \frac{1}{q})^{-1}<l < \infty \) we get
where \(C= C(n,s,p,q,l) \), \(\vartheta = 2-\frac{1}{r}- \frac{1}{q} - \frac{p}{lr^\prime }\) and \(\nu = 1-\frac{1}{r} - \frac{1}{q}\).
Theorem 1.2
Let \(\Omega \subset {\mathbb {R}}^n\) be a bounded and open set, \(I=(t_0,t_1]\), \(p\ge 2\), \(0<s<1\). Consider q and r such that
In addition, assume that \(r\ge p^\prime \),
Define the exponent
Suppose u is a local weak solution of
such that
Then
More precisely, given \(\alpha < \Theta \) satisfying
for every \(R>0\), \(x_0\in \Omega \) and \(T_0\) such that
there exists a constant \(C=C(n,s,p, q,r,\alpha )>0\) such that
for any \((x_1,t_1),\,(x_2,t_2)\in Q_{R/2,(R/2)^{s\,p}}(x_0,T_0)\), with
1.1 Known results
Recently, there has been a growing interest in nonlocal problems of both elliptic and parabolic types. For studies of fractional p-Laplace operators with different (continuous) kernels, see [4]. Parabolic equations of the type (1.1) were first considered in [42] with a slightly different diffusion operator. See also [1, 39, 48] and [49] for studies of the existence, uniqueness and long time behavior of solutions.
A noteworthy area of investigation has been devoted to adapting the classical De Giorgi–Nash–Moser theory for nonlocal equations. Local boundedness, Hölder estimates and Harnack inequalities have been established in the elliptic case under general assumptions on the kernels; see, for instance, [19, 23, 24, 32].
Here we seize the opportunity to mention [16,17,18] and [51] which contain regularity results for parabolic nonlocal equations.
Local boundedness for parabolic nonlocal equations has been studied, for instance, in [11, 22, 33, 45]. In particular, the local boundedness of the solutions to equations modeled on (1.1) with zero right-hand side was obtained in [45]. The results concern operators of the form
where K is a measurable kernel, which is symmetric in the space variables and satisfies the ellipticity condition
Later in [22], local boundedness for certain right-hand sides of the form f(x, t, u) was established. See also [3] for a recent boundedness result in the setting of nonlocal kinetic Kolmogorov–Fokker–Planck equations. All the aforementioned local boundedness results have a particular unnatural assumption, \(u\in L^\infty (I;L_{sp}^{p-1}({\mathbb {R}}^n))\). It is more natural to assume \(u\in L^{p-1}(I;L_{sp}^{p-1}({\mathbb {R}}^n))\). This difficulty has been completely resolved in [34] when \(p=2\) and generalizes to the nonlinear setting in [10].
[46] contains a Harnack inequality for nonlinear parabolic equations with zero right-hand side, see also [34] for a full Harnack inequality with optimal tail assumption for \(p=2\). Hölder regularity has also been established in [13, 27] for \(p=2\) and for locally bounded solutions in [2] and [37] for all \(1<p<\infty \) for equations with zero right-hand side.
The question of higher regularity of solutions to nonlocal equations has also been a subject of intensive study during the past few years. For instance, see [28, 43] for a nonlocal Schauder-type theory. We also refer to [14, 15] for nonlocal analogs of Krylov–Safanov and Evans–Krylov theorems. We refer to [6, 7, 11, 12, 26, 40, 41] for studies of higher regularity in the variational setting. In particular, in [7] they prove Hölder continuity of the solutions to (1.1) with explicit exponents (for \(f=0\) and \(K= |x-y|^{-n-sp}\)). Recently in [29], the same type of result has been established for nonlocal equations with double phase that is for diffusion operators involving two different degrees of homogeneity and differentiability.
Perturbative arguments have been very successful in obtaining sharp boundedness and Hölder regularity estimate at least in the elliptic setting, see, for instance, [25, 35]. See also [36] for an overview of the local theory. In this study, continuing the work in [7], we perform a perturbative argument to obtain Hölder continuity estimates with explicit exponents for equations with a right-hand side. However, we have to say that the current work has some unnatural assumptions that have yet to be overcome.
1.1.1 Discussion of the results and comparison to some previous works
Our results contain an unnatural assumption \(r\ge p^\prime \), as well as the assumption \(u \in L^p(I;L_{sp}^{p-1}({\mathbb {R}}^n))\) in Theorem 1.1. We use these assumptions in two places. First and foremost these assumptions are needed to ensure the existence of a solution to (3.1), the so-called (s, p)-caloric replacement of our solution. This limitation comes from the regularity assumption on the boundary condition in Theorem 2.12 which is essentially the same as [7, Theorem A.3] see Remark 2.13. We also use the assumption \(r\ge p^\prime \) in obtaining the estimates in Lemma 3.2. We believe it is possible to overcome this issue by an interpolation argument, see Remark 3.3. It has to be mentioned that we also use the assumption \(u \in L^p(I;L_{sp}^{p-1}({\mathbb {R}}^n))\) to justify testing the equation with powers of the solution in Appendix B. Having said this, it is reasonable to expect Theorem 1.1 to hold for any weak solution under the assumption
as this is the only assumption that appears in the estimates. The same extra assumptions on q and r are present in Theorem 3.6 due to the same reason as in Theorem 1.1. Furthermore, we assume our solutions to have bounded tail in time, that is, \(u\in L^\infty (I; L^{p-1}_{sp}({\mathbb {R}}^n))\). In light of the recent developments in [10, 34], one can actually weaken the assumptions on the tail. In particular, by using [10, Theorem 1.2] instead of [7, Theorem 1.2] in the proof of Theorem 3.6, with some small modifications in the argument one should be able to obtain the Hölder continuity of the solutions under the assumption \(u \in L^p(I;L^{p-1}_{sp}({\mathbb {R}}^n))\), and the same assumptions on q, r as in Theorem 3.6. We also believe that it is possible to avoid using Proposition 3.4 in the proof of Theorem 3.6, by using [10, Theorem 1.1 and Theorem 1.2]. As improving upon this assumption does not improve our main result, Theorem 1.2, we do not complicate the article by going through the details of this issue. Furthermore, we actually expect the result to be true under the weaker assumption \(u \in L^l(I;L^{p-1}_{sp}({\mathbb {R}}^n))\) for some \(l>p-1\) and without the assumption \(r\ge p^\prime \), but the current restrictions in the article especially with respect to the existence of the (s, p)-caloric replacement do not allow us to obtain such a result.
Let us also mention that the local boundedness and Hölder regularity results mentioned above hold for a more general class of equations with measurable coefficient \(u_t + L_k u =0\), where \(L_k\) is as in 1.4. Although we write our results for the equation \(u_t + (-\Delta _p)^s u = f\), the arguments in the proofs of Theorem 1.1 and Theorem 3.6 can be adapted to the equations \(u_t + L_k u = f\) with measurable, asymmetric, uniformly elliptic coefficients easily. The only difference is that a dependence on the ellipticity coefficients will appear in the constants. But the question of what assumption is needed on the kernel to get higher Hölder regularity is subtle. We refer to [11, 25, 40] for a study of this issue.
The equation \(u_t -\Delta _p u = f\) can be seen as a limit of the equation \(u_t +(1-s)c(n,p)(-\Delta _p)^s u = f\) as \(s \nearrow 1\). A relevant question is whether the estimates provided here in the article are stable with respect to s as \(s \nearrow 1\). We have to admit that we did not keep track of the dependence of the constants on s while writing this article, and we wrote the article for the operator \(\partial _t + (-\Delta _p)^s\) instead of \({\partial _t + (1-s) (-\Delta _p)^s}\). Still, we can say a few words on the dependence of our constants on s for those who might be interested in pursuing this question. The proofs of Theorem 1.1 and Theorem 3.6 are combinations of local boundedness estimates in [10, Theorem 1.1] and the Holder continuity result [7, Theorem 1.2] for the equations with zero right-hand side, together with the comparison estimates of Lemma 3.2 and Proposition 3.4. [7, Theorem 1.2] is stable as \(s \nearrow 1\) see [7, Remark 1.7], as for [10, Theorem 1.1] they did not specify the dependence of their constants on s in their article. In Lemma 3.2 and Proposition 3.4, the dependence of the constants on s comes from the Sobolev and Morrey inequalities. The constants in these inequalities behave like \(s(1-s)\) with respect to s, but it has to be mentioned that we update the constants to be greater than one in several places. It might be the case that if one considers the operator \(\partial _t +(1-s)(-\Delta _p)^s\) instead, the estimates in Lemma 3.2 and Proposition 3.4 would become robust as \(s \nearrow 1\). We cannot specify the dependence of the constant in Theorem 1.2 on s specifically. The main difficulty lies in the proof of Lemma 3.7, which is proved by contradiction.
Now we compare the main results of the article to some other works.
Local boundedness and continuity In the recent work [11], they address the issue of local boundedness when \(p=2\) for a more general class of operators by a direct proof. By avoiding the difficulty of the existence of the caloric replacement, their result does not contain the extra assumption \(r\ge p^\prime \), although they assume \(u \in L^\infty (I;L_{sp}^{p-1}({\mathbb {R}}^n))\).
We compare our boundedness result to [22]. Their result concerns more general right-hand sides depending on the solution as well. In the limiting case of \(s \rightarrow 1\), they reproduce the local boundedness result contained in [21] for the evolution p-Laplacian equation. To compare the results, if we restrict their result to right-hand sides that are u-independent, their assumption on the integrability becomes \(q,r >\frac{n+sp}{sp}(\frac{p(n+2\,s)}{2sp +(p-1)n}) \). Their analysis is done with the same integrability assumption in time and space. Our local boundedness result, Theorem 1.1, contains this range of exponents.
In the limiting case when s goes to 1, 1.5 become \(1-\frac{1}{r} - \frac{n}{pq} >0\). This is in accordance with the classical condition for boundedness of the evolution p-Laplace equation, see, for example, Remark 1 in [38], there they have a finer assumption formulated in terms of the Lorentz norm of the right-hand side, and moreover, they obtained estimates in terms of a parabolic version of Wolf potentials. It would be interesting to obtain finer estimates beyond \(L^p\) spaces, although we do not pursue this question in this article. If we assume the same integrability in time and space, the condition \(1-\frac{1}{r} -\frac{n}{spq}>0 \) reduces to \(f\in L^{{\hat{q}}}\) with \({\hat{q}}>\frac{n+p}{p}\). This matches the condition in [50].
Now we turn our attention to the nonlocal elliptic (time-independent) case. For \(r=\infty \), the condition for boundedness and basic Hölder continuity becomes
In the case \(sp<n\), this is the same condition for local boundedness and continuity contained in [8, 35]. When \(sp>n\) and \(q\ge 1\), the boundedness and Hölder continuity for the time-independent equation is automatic using Morrey’s inequality. The question of whether the solutions are locally bounded under the equality case of (1.5) is subtle. On the one hand, if \(r=\infty \) even in the time-independent (elliptic) setting one requires the strict inequality \(q> \frac{n}{sp}\) to obtain boundedness; on the other hand, local boundedness is obtained in the case \(r=1\) and \(q=\infty \) in [34], see also [10].
There are actually local boundedness and Hölder continuity results available for the equations with zero right-hand side if \(p< 2\). One could try to prove local boundedness and basic Hölder regularity of the solutions for the solutions of the equations with right-hand side in the singular case \(p<2\) as well. We have to warn the reader that some of the arguments in this article do not carry over to the singular case as they are written here. We use the condition \(p\ge 2\) extensively, in particular in the Pointwise inequalities (2.1) and (2.3). We feel that it is better if we leave the study of the singular case to another work. We also have to mention that if one is only interested in local boundedness estimates, doing a nonperturbative argument is more suitable, as one can also deal with sub- and supersolutions.
Hölder continuity exponent: In the case \(r= \infty \), the critical Hölder continuity exponent
reduces to \(\min { \lbrace \Theta ,\frac{sp}{p-1}(1-\frac{n}{spq})\rbrace } \) which matches the results in [6]. Although the results reported in [6] require a strict inequality \(\alpha < \min { \lbrace \Theta ,\frac{sp}{p-1}(1-\frac{n}{spq})\rbrace }\), an inspection of the proofs reveals that the strict inequality is only needed when the minimum corresponds to \(\Theta \). The assumptions needed for their proof are actually \(\alpha \le \frac{sp}{p-1}(1-\frac{n}{spq})\) and \(\alpha < \Theta \). Through a finer estimate in [25], they have addressed this issue further and proved that given \(\alpha \le \Theta \), if the right-hand side f belongs to the Marcinkiewicz space \( L^{\frac{n}{sp-\alpha (p-1)}, \infty }(\Omega )\) then the solution is \(C^\alpha _{\textrm{loc}}(\Omega )\).
Let us also compare our results to the local p-parabolic equation studied in [47] where precise Hölder continuity exponents are obtained. If we send s to 1, (1.6) becomes
which is in accordance with the result in [47].
In [29], explicit Hölder continuity exponents for the more general case of double phase nonlocal diffusion operators were obtained. The ideas explored there are similar to the ones in [7], but their result allows for a bounded right-hand side instead of just zero. Their result implies the Hölder continuity exponent that we get in the case of \(f\in L^\infty \), although with a slightly different estimate of the Hölder constants. Let us also mention that in the recent work [11] the conclusions of Theorem 1.2 have been obtained when \(p=2\), for a more general class of operators and kernels. Although [11, Theorem 1.2] does not contain the extra assumption \(r\ge p^\prime \), their argument is similar to our proof of Theorem 1.2 and the same difficulty regarding the existence of the (s, p)-caloric replacement is present in their proof. This difficulty has not been addressed properly in their article. In the assumptions [11, (A.1)] for their existence theorem, the regularity assumption \(\xi _t \in L^2((0,T);W^{s,2}(\Omega ^\prime ))^\star \) is present. We are not able to verify this assumption when \(\xi \) is a solution of
for \(f \in L^{q,r}\) such that \(\frac{n}{2sq}+ \frac{1}{r} <1\) as it is claimed by [11, Remark 6]. See Remark 2.13 for a possible strategy for resolving this issue.
Let us close this section with the question of how much regularity one should expect if the solution has a lower integrability of the tail in time. Namely given a weak solution u of the equation
such that \(u \in L^{l}(L_{sp}^{p-1}({\mathbb {R}}^n))\) how much Hölder regularity does the solution have. Let us first mention that an example in [34, Example 5.2] shows that the assumption \(u \in L^{p-1}(L_{sp}^{p-1}({\mathbb {R}}^n))\) does not ensure the Hölder regularity of the solution. On the other hand, it is proved in [34] and [10] that if \(l>p-1\) then the solution is Hölder continuous, and the general strategy in these works is to treat the following nonlocal term
which appears in the Caccioppoli inequalities, as a right-hand side in \(L^{\frac{l}{p-1}}(I; L^\infty (B))\). See [10, Section 1.2] for more details. Following this general philosophy, one can expect the solution to be \(C^\alpha _x\) and \(C^{\frac{\alpha }{sp- (p-2)\alpha }}_t\) with
However, at the moment we do not have a definite answer to how this can be shown rigorously.
1.2 Plan of the paper
In Sect. 2, we introduce some notations and preliminary lemmas. We also restate and adapt a result on the existence of solutions to our setting.
In Sect. 3, we establish basic local Hölder regularity and boundedness for local weak solutions.
Section 4 is devoted to proving Theorem 1.2. A so-called tangential analysis is performed to get specific Hölder continuity exponents in terms of q, r, s and p.
The article is also accompanied by two appendices. In the first one, Appendix A, we work out the details for a modified version of [7, Theorem 1.1]. The aim is to bound the Hölder seminorm of the solution in terms of the tail quantity.
In Appendix B, we justify using certain test functions in the weak formulation of (1.1).
2 Preliminaries
2.1 Notation
We define the monotone function \(J_p: {\mathbb {R}}\rightarrow {\mathbb {R}}\) by
We use the notation \(B_R(x_0)\) for the open ball of radius R centered at \(x_0\). If the center is the origin, we simply write \(B_R\). We use the notation of \(\omega _n\) for the surface area of the unit n-dimensional ball. For parabolic cylinders, we use the notation \(Q_{r,T}(x_0,t_0):= B_r(x_0)\times (t_0-T, t_0]\). If the center is the origin, we write \(Q_{r,T}\).
We will work with the fractional Sobolev space extensively:
where the seminorm \([\psi ]_{W^{s,p}({\mathbb {R}}^n)}\) is defined as below
We also need the space \(W^{\beta , q }(\Omega )\) for a subset \(\Omega \subset {\mathbb {R}}^n\), defined by
where
In the following, we assume that \(\Omega \subset {\mathbb {R}}^n\) is a bounded open set in \({\mathbb {R}}^n\). We define the space of Sobolev functions taking boundary values \(g \in L^{q-1}_{sp}({\mathbb {R}}^n) \) by
where \(\Omega ^\prime \) is an open set such that \(\Omega \Subset \Omega ^\prime \).
We recall the definition of tail space
which is endowed with the norm
For every \(x_0\in {\mathbb {R}}^n\), \(R>0\) and \(u\in L^q_{\alpha }({\mathbb {R}}^n)\), the following quantity
plays an important role in regularity estimates for solutions to fractional problems.
Let \(I \subset {\mathbb {R}}\) be an interval and let V be a separable, reflexive, Banach space endowed with a norm \(\Vert \,\bullet \,\Vert _V\). We denote by \(V^\star \) its topological dual space. Suppose that v is a mapping such that for almost every \(t \in I\), we have \(v(t) \in V\). If the function \(t \rightarrow \Vert v(t)\Vert _V\) is measurable on I and \(1 \le p \le \infty \), then v is an element of the Banach space \(L^p(I; V)\) if and only if
By [44, Theorem 1.5], the dual space of \(L^p(I; V)\) can be characterized according to \((L^p(I; V ))^\star = L^{p^\prime }(I; V^\star ).\) We write \(v \in C(I; V )\) if the mapping \(t \rightarrow v(t)\) is continuous with respect to the norm on V.
2.2 Pointwise inequalities
We will need the following pointwise inequality: Let \(p \ge 2\), then for every \(A,B \in {\mathbb {R}}\) we have
For a proof look at [7, Remark A.4], a close inspection of the proof reveals that the constant can be taken as \(C= 3 \cdot 2^{p-1}\). Before stating the next inequality, we recall [8, Lemma A.2].
Lemma 2.1
Let \(1< p < \infty \) and \(g:{\mathbb {R}}\rightarrow {\mathbb {R}}\) be an increasing function, and define
Then
Lemma 2.2
For \(p\ge 2\) and \(\beta \ge 1\),
where \((t)^{+}_M:=\min {\lbrace \max {\lbrace t,0\rbrace },M\rbrace }\).
Proof
We consider three cases according to the sign of \(a-b-c+d\). If \(a-b-c+d=0\) both the left-hand side and the right-hand side of (2.2) are zero. Now we verify the inequality for \(a-b-c+d > 0\)
First notice that using (2.1) with \(A= a-b \; \text {and}\; B= c-d\):
using the fact that \(a-b -c+d > 0\), we arrive at
Now we use Lemma 2.1 with \(g(t)= ((t)^{+}_M +\delta )^\beta \). Then with \(G= \int _0^t g^\prime (\tau )^{\frac{1}{p}} \;\textrm{d}\tau \),
By Lemma 2.1,
Hence,
Using (2.3) in the above inequality concludes the proof. It only remains to verify the case \(a-b-c+d <0\), now we are in the previous position and can use with (b, a, d, c) instead of (a, b, c, d) to obtain
As
and
we obtain (2.2) \(\square \)
The following pointwise inequality is a direct consequence of the convexity of the mapping \(t \rightarrow |t|^{\alpha }\) for \(\alpha \ge 1\).
2.3 Functional inequalities
We need the following basic inequalities for the tail.
Lemma 2.3
Let \(\alpha >0\), \(1\le q< \infty \), and \(u,\,v \in L_\alpha ^q({\mathbb {R}}^n)\) such that \(u=v\) on \({\mathbb {R}}^n \setminus B_R(x_0)\). Then for any \(\sigma <1\),
Proof
\(\square \)
For a proof of the following result, see [6, Lemma 2.3].
Lemma 2.4
Let \(\alpha >0\), \(0<q<\infty \). Suppose that \(B_r(x_0) \subset B_R(x_1)\). Then for every \(u\in L^q_\alpha ({\mathbb {R}}^n)\), we have
If in addition \(u \in L^m_{\textrm{loc}}({\mathbb {R}}^n)\) for some \(q <m \le \infty \), then
where \(\omega _n\) is the measure of the n-dimensional open ball of radius 1.
We also recall the following Sobolev- and Morrey-type inequalities:
Proposition 2.5
Suppose \(1<p<\infty \) and \(0<s<1\). Let \(\Omega \subset {\mathbb {R}}^n\) be an open and bounded set. Define \(p_s^\star \) as
For every \(u \in W^{s,p}({\mathbb {R}}^n)\) vanishing almost everywhere in \({\mathbb {R}}^n {\setminus } \Omega \), we have
In particular, the following Poincaré inequality holds true
for some \(C=C(n,s,p)\). Furthermore, in the supercritical range of exponents functions in \(W^{s,p}({\mathbb {R}}^n)\) are Hölder continuous and the following inequality holds true:
Remark 2.6
The above Sobolev-type inequalities are also valid for functions \(u \in X_0^{s,p}(\Omega , \Omega ^\prime )\), where \(\Omega \) is a bounded open set and \(\Omega ^\prime \) is an open set such that \(\Omega \Subset \Omega ^\prime \). This can be seen using the fact that there is an extension domain containing \(\Omega \) and included in \(\Omega ^\prime \).
We will often use the following special application of Hölder’s inequality
where \(q_1<q_2 \,, r_1\le r_2\). The following interpolation inequality (see, e.g., [5]) will be useful.
Lemma 2.7
If w is contained in \(L^{q_1,r_1}(\Omega \times J) \cap L^{q_2, r_2}(\Omega \times J)\), then w is contained in \(L^{{\tilde{q}}, {\tilde{r}}}(\Omega \times J)\), where
Moreover,
The following three lemmas will be needed in the proof of our local boundedness result (Proposition 3.4).
Lemma 2.8
Let \(sp \ne n\) and assume that w is in \(L^{p}\bigl ( (T_0-R^{sp},T_0);W^{s,p}({\mathbb {R}}^n)\bigr ) \cap L^{p,\infty }(Q_{R,R^{sp}}(x_0,T_0))\) and w(x, t) is zero for all \(x \in {\mathbb {R}}^n {\setminus } B_R(x_0)\), for almost every \(t \in (T_0-R^{sp},T_0]\). Then w is in \(L^{p q^\prime , p r^\prime } (Q_{R,R^{sp}}(x_0,T_0))\) as long as q, r satisfy
Moreover,
where C depends on \(n,s \text { and } p\). In particular, in the case of \(\frac{1}{r} + \frac{n}{spq}=1\) we have
Proof
Consider a pair of exponents \({\tilde{r}} = (\frac{1}{r^\prime } - (1-\frac{1}{r} - \frac{n}{spq}))^{-1}= \frac{spq}{n}\), and \({\tilde{q}} = q^\prime \) such that \(\frac{1}{{\tilde{r}}^{\, \prime }} + \frac{n}{sp{\tilde{q}}^{\, \prime }} =1 \). Using Hölder’s inequality (2.11), we obtain
Now we split the proof into two cases depending on whether \(sp<n\) or not.
Case \(sp<n\): We use Lemma 2.7 with the choice
This yields
The above relations hold for \(\lambda = \frac{1}{{\tilde{r}}} = \frac{n}{sp{\tilde{q}}^{\, \prime }}\), and using Sobolev’s inequality 2.6, we arrive at
By using Young’s inequality, we get
Case \(sp>n\): In this case, we use the following interpolation between Hölder and \(L^p\) spaces:
See [9, Lemma 2.2] for a proof. In light of the Morrey-type inequality (2.10), for almost every \(t \in (T_0-R^{sp}, T_0) \) we arrive at
Now we interpolate once more between \(L^p\) and \(L^\infty \) to obtain
We raise both sides to the power \(p{\tilde{r}}\) and integrate with respect to t. Recalling that \(\frac{1}{{\tilde{r}}}=\frac{n}{sp {\tilde{q}}^\prime }\), we obtain
Taking the \({\tilde{r}}\) root and applying Young’s inequality, we obtain the desired estimate:
\(\square \)
Lemma 2.9
Let \(sp=n,\, q\ge 1, \text { and } r\ge 1 \) such that
Assume that \(w \in L^{l,p}(Q_{R,R^{sp}}(x_0,T_0))\cap L^{p,\infty }(Q_{R,R^{sp}}(x_0,T_0))\) for some l such that
Then w belongs to \(L^{pq^\prime , p r^\prime }(Q_{R,R^{sp}}(x_0,T_0))\) and
Proof
We use Lemma 2.7 with the choice
Due to the assumption \(\frac{1}{l}= \frac{r^\prime }{p}(1-\frac{1}{r}-\frac{1}{q})\), the above equalities hold for \(\lambda = \frac{1}{r^\prime }\). Hence, we get
Therefore, recalling that \(\lambda = \frac{1}{r^\prime }\)
Using Young’s inequality for the right-hand side, we can conclude
\(\square \)
2.4 Weak solutions
Definition 2.10
For any \(t_0,t_1\in {\mathbb {R}}\) with \(t_0<t_1\), we define \(I=(t_0,t_1]\). Let
for any open \({\mathcal {K}}\) such that \({\mathcal {K}}\Subset \Omega \). We say that u is a local weak solution to the equation
if for any closed interval \(J=[T_0,T_1]\subset I\), the function u is such that
and it satisfies
for any \(\varphi \in L^p(J;W^{s,p}(\Omega ))\cap C^1(J;L^2(\Omega ))\) which has spatial support compactly contained in \(\Omega \). In equation (2.13), the symbol \(\langle \,\bullet \,,\,\bullet \,\rangle \) stands for the duality pairing between \(W^{s,p}(\Omega )\) and its dual space \((W^{s,p}(\Omega ))^*\).
Now, we define the notion of a weak solution to an initial boundary value problem.
Definition 2.11
Let \(I=[t_0,t_1]\), \(p\ge 2\), \(0<s<1\), and \(\Omega \Subset \Omega ^\prime \), where \(\Omega ^\prime \) is a bounded open set in \({\mathbb {R}}^n\). Assume that the functions \(u_0,f\) and g satisfy
We say that u is a weak solution of the initial boundary value problem
if the following properties are verified:
-
\(u\in L^p(I;W^{s,p}(\Omega '))\cap L^{p-1}(I;L_{sp}^{p-1}({\mathbb {R}}^n))\cap C(I;L^2(\Omega ))\);
-
\(u\in X_{{\textbf{g}}(t)}(\Omega ,\Omega ')\) for almost every \(t\in I\), where \(({\textbf{g}}(t))(x)=g(x,t)\);
-
\(\lim _{t\rightarrow t_0}\Vert u(\,\bullet \,,t) - u_0\Vert _{L^2(\Omega )}=0\);
-
for every \(J=[T_0,T_1]\subset I\) and every \(\varphi \in L^{p}(J;X_0^{s,p}(\Omega ,\Omega '))\cap C^1(J;L^2(\Omega ))\)
$$\begin{aligned} \begin{aligned}&\quad -\int _J\int _\Omega u(x,t)\,\partial _t\varphi (x,t)\;\textrm{d}x\;\textrm{d}t\\&\quad + \int _J\iint _{{\mathbb {R}}^n\times {\mathbb {R}}^n}\frac{J_p(u(x,t)-u(y,t))\,(\varphi (x,t)-\varphi (y,t))}{|x-y|^{n+sp}}\;\textrm{d}x\;\textrm{d}y\;\textrm{d}t \\&= \int _\Omega u(x,T_0)\,\varphi (x,T_0)\;\textrm{d}x -\int _\Omega u(x,T_1)\,\varphi (x,T_1)\;\textrm{d}x \\&\quad + \langle f,\varphi \rangle . \end{aligned} \end{aligned}$$
Let us mention that given a local weak solution in a cylinder \(I \times \Omega ^\prime \), where \(I=(t_0,t_1]\) and \(\Omega ^\prime \) is a bounded, open subset of \({\mathbb {R}}^n\), by considering a smaller cylinder \(J \times \Omega \) such that \(\Omega \Subset \Omega ^\prime \) and J is a closed interval compactly contained in I we end up a weak solution in the smaller cylinder \(J \times \Omega \).
Throughout the article, we work with right-hand sides \(f\in L^{p^\prime } (I;L^{(p^\star _s)^\prime }(\Omega ))\), where \(p_s^\star \) is the Sobolev exponent and we consider it to be infinity if \(sp>n\). An application of Hölder’s inequality together with the Sobolev–Morrey inequalities ensures that \(f\in f\in L^{p'}(I;(X_0^{s,p}(\Omega ,\Omega '))^*) \subset \left( L^p(I;X^{s,p}_0 (\Omega \,, \Omega ^\prime ))\cap L^\infty (I;L^2(\Omega ))\right) ^{\star }\) with the duality pairing
Theorem 2.12
Let \(p\ge 2\), let \(I = (T_0,T_1]\) and suppose that g satisfies
Suppose also that
Then for any initial datum \(g_0\in L^2(\Omega )\), there exists a unique weak solution u to problem
Proof
In [7, Theorem A.3], the same result is proved with a stronger condition \(g_t \in L^{p^\prime }(I;W^{s,p}(\Omega ^\prime )^\star )\). The stronger condition is not needed in the proof. This condition can be replaced with \(g_t \in L^{p^\prime }(I,X^{s,p}_0 (\Omega ; \Omega ^\prime )^\star )\) in all of the steps in the proof, except that the construction gives us a \(C(I;L^2(\Omega ))\) solution. There, the stronger assumption is used only to show that the boundary condition is in \(C(I;L^2(\Omega ))\), which we assume here. \(\square \)
Remark 2.13
The condition \(\partial _t g \in L^{p'}(I;(X_0^{s,p}(\Omega ,\Omega '))^*)\) is too strong. This condition forces us to assume \(r \ge p^\prime \), \(q \ge (p_s^\star )^\prime \) and \(u \in L^p(I; L^{p-1}_{sp}({\mathbb {R}}^n))\) in Proposition 3.1 and hence in all our results. A more natural condition would be to assume \(\partial _t g \in \left( L^p(I;X^{s,p}_0 (\Omega \,, \Omega ^\prime ))\cap L^\infty (I;L^2(\Omega ))\right) ^{\star }\). We believe it is possible to overcome this difficulty by pursuing an approximation procedure in the spirit of [35, Theorem 1.1 and Lemma 4.1].
3 Basic Hölder regularity and stability
Throughout the rest of the article, we assume \(0<s<1\) and \(2\le p < \infty \).
Here, we argue that the norm of the (s, p)-caloric replacement of u is close to u if f is small enough. By the (s, p)-caloric replacement of u in a cylinder \(B_\rho (x_0) \times I\), we mean the solution to the following
Here \(\tau _0\) is the initial point of the interval I. First we show the existence of a (s, p)-caloric replacement using Theorem 2.12
Proposition 3.1
Let u be a local weak solution of \( u_t + (-\Delta _p)^s u=f \) in the cylinder \(B_\sigma \times J\), for some interval \(J=(t_1,t_2]\) with \(f \in L^{q,r}_{\textrm{loc}}(B_\sigma \times J)\) such that \(r\ge p^\prime \),
In addition, we assume that \(u \in L^p(J;L_{sp}^{p-1}(R^n))\). Then for any \(0<\rho < \sigma \), and closed interval \(I \Subset J\), the (s, p)-caloric replacement of u in \(B_\rho (x_0)\times I\) (weak solution to (3.1)) exists.
Proof
We shall check the conditions in Theorem 2.12. If they are satisfied, there exists a unique weak solution \(v \in L^p(I, W^{s,p}(B_{\sigma })) \cap L^{p}(I;L_{sp}^{p-1}({\mathbb {R}}^n)) \cap C(I;L^2(B_\rho ))\) to the problem (3.1). The only condition on u that is not immediate from the fact that u is weak solution is \(\partial _t u \in L^{p^\prime }(I;X^{s,p}_0 (B_\rho \,, B_{\sigma })^\star ) \). We have to show that for every function \(\psi \in L^p(I;X^{s,p}_0 (B_\rho \,, B_{\sigma }))\)
Here we only write the proof for the case \(sp< n\), the case of \(sp \ge n\) is similar, except that one has to use the critical case of Sobolev inequality and the Morrey inequality instead of using the Sobolev inequality. We shall verify (3.2) for test functions belonging to the dense subspace, \(\psi \in L^p(I;X^{s,p}_0 (B_\rho \,, B_{\sigma }))\cap C^1_{0}(I; L^2(B))\). We use the equation to do so. We have
By Hölder’s inequality, we have
For the other nonlocal term, we note that for every \(x \in B_\rho \) and \(y \in {\mathbb {R}}^n\setminus B_{\sigma }\) we have \(|y| \le \frac{\sigma }{\sigma -\rho } |x-y|\). Hence,
Therefore,
By Hölder’s inequality, we have
For the other term,
Since \(f \in L^{p^\prime }(I; L^{(p_s^\star )^\prime }(B_\rho ))\), by Hölder’s inequality and Sobolev’s inequality we obtain
Therefore, combining with (3.3), (3.4), and (3.5) we obtain
\(\square \)
Lemma 3.2
Assume that \(f\in L^{q,r}_{\textrm{loc}}(Q_{\sigma ,\sigma ^{sp}}(x_0,T_0))\) with \(r\ge p^\prime \),
Let u be a local weak solution of \(\partial _t u + (-\Delta _p)^s u= f\) in \(Q_{\sigma ,\sigma ^{sp}}(x_0,T_0)\), such that \(u \in L^p_{\textrm{loc}}((T_0-\sigma ^{sp},T_0]; L^{p-1}_{sp}({\mathbb {R}}^n))\). Let \(\rho < \sigma \) and consider v to be the (s, p)-caloric replacement of u in \(Q_{\rho ,\rho ^{sp}}(x_0,T_0)\). Then we have
and
with \(\xi = spp^\prime (1- \frac{1}{r} - \frac{n}{spq})\) and \(C= C(n,s,p)\), in the case \(sp\ne n\). In the case \(sp=n\), we can take \(\xi = spp^\prime (1 - \frac{1}{r} - \frac{1}{q})\), with \(C=C(n,s,p,q)\) also depending on q.
Proof
Let \(J:=[T_0-\rho ^{sp},T_0]\), throughout the proof, we drop the dependence of the balls on the center and write \(B_\rho \) instead of \(B_\rho (x_0)\), and \(Q_{\rho ,\rho ^{sp}}\) instead of \(Q_{\rho ,\rho ^{sp}}(x_0,T_0)\).
By subtracting the weak formulation of the equations (2.13) for u and v with the same test function \(\varphi (x,t) \in L^{p}(J;X_0^{s,p}(B_{\rho },B_{\sigma }))\cap C^1(J;L^2(B_\rho ))\), we get
Now we take \(\varphi := u-v\), which belongs to \(L^p(J; X_0^{s,p}(B_{\rho };B_\sigma ))\), but it may not be in \(C^1(J;L^2(B_\rho ))\). We justify taking this as a test function in Appendix B. By Proposition 6.1 with \(F(t)=t\), we get
where in the third line we have used \(u(x,T_0-\rho ^{sp}) = v(x,T_0-\rho ^{sp})\). The left-hand side is essentially the \(W^{s,\,p}\) seminorm. By the pointwise inequality (2.1),
Therefore, by (3.9) and Hölder’s inequality
Now we consider three cases: \(sp<n\), \(sp>n\) and \(sp=n\).
Case \(sp < n\). By Hölder’s inequality (2.11) and Sobolev’s inequality (2.6), we have
Combined with (3.10), this yields
where \(C= C(n,s,p)\). By the Poincaré inequality,
Also from (3.12) and (3.11), we get
Case \(sp>n\). In this case, we use Morrey’s inequality (2.7) and Hölder’s inequality and obtain
Together with (3.10), this implies
By the Poincaré inequality,
Combining (3.13) and (3.14), we get
Case \(sp=n\). In this case, we use the critical case of Sobolev’s inequality (2.8) for \( l= q^\prime \) and obtain
Hence, using Hölder’s inequality, we have for any \(r\ge p^\prime \)
The above constant \(C=C(n,s,p,q)\) does blow up as q goes to 1. In a similar way as in the prior cases, we get for \(q>1\) and \(r\ge p^\prime \)
and
Using that \(|B_\rho |\sim \rho ^{n}\) and \(|I| \sim \rho ^{sp}\), we can conclude that
and
Here in the case of \(sp\ne n\),
and in the case \(sp=n\),
\(\square \)
Remark 3.3
In Lemma 3.2, we assume the same conditions as in Proposition 3.1. These assumptions are used in the proof not only to ensure the existence of the (s, p)-caloric replacement but also to derive (3.11) and (3.13). As mentioned in Remark 2.13, one can expect the existence of the (s, p)-caloric replacement under a more general condition for the right-hand side. If such an existence theorem is available, one can expect the estimates in Lemma 3.2 to hold true for more general right-hand sides. In the proof of Lemma 3.2, we only used the diffusion term in (3.9), but the stronger estimate
holds true. It might be possible to utilize an interpolation argument similar to Lemma 2.8 to replace the equations (3.11) and (3.13) and relax the assumptions on q and r. See also [12, Lemma 2.2]. However, the nonhomogeneity of the equation is for sure a challenge in pursuing this line of reasoning.
Next, we perform a Moser iteration to get an \(L^\infty \) bound for the difference between the solution and its (s, p)-caloric replacement.
Proposition 3.4
Let u be a local weak solution of
with \(f \in L^{q,r}_{\textrm{loc}} (Q_{\sigma ,\sigma ^{sp}}(x_0,T_0))\) such that
In addition, assume that \(u\in L^p_{\textrm{loc}} \bigl ( (T_0-\sigma ^{sp},T_0];L_{sp}^{p-1}({\mathbb {R}}^n)\bigr )\), \(r\ge p^\prime \),
Let v be the (s, p)-caloric replacement of u in \(Q_{R,R^{s p}}(x_0,T_0)\), with \(R< \sigma \). Then in the case of \(sp \ne n\), we have
where \(\nu = 1-\frac{1}{r} - \frac{n}{spq}\) and
In the case of \(sp=n\), given any l such that \( \frac{p}{r^\prime }(1-\frac{1}{r} - \frac{1}{q})^{-1}<l < \infty \) we get
where \(\vartheta = 2-\frac{1}{r}- \frac{1}{q} - \frac{p}{lr^\prime }\) and \(\nu = 1-\frac{1}{r} - \frac{1}{q}\).
Proof
Throughout the proof, we write \(Q_{R,R^{sp}}\) instead of \(Q_{R,R^{s,p}}(x_0,T_0)\) and \(B_R\) instead of \(B_R(x_0)\). We also define the interval J to be \(J:= (T_0-R^{sp},T_0]\). First, we verify that our assumptions ensure that the (s, p)-caloric replacement of u exists. If \(sp \ge n\), we have explicitly assumed what is needed to use Proposition 3.1. If \(sp< n\), we have to verify that \(q\ge (p_s^\star )^\prime \). This follows from the assumption \(\frac{1}{r} + \frac{n}{spq} < 1\). Indeed
and it is straightforward to verify that \(\frac{n}{sp}\ge (p_s^\star )^\prime \). This shows that v, the (s, p)-caloric replacement of u exist. Let us also mention that the assumptions in Lemma 3.2 are the same as in Proposition 3.1, and we can use this lemma. Now, we test the equations with powers of \(u-v\) and perform a Moser iteration. Using Proposition 6.1 with
and
we get
In the last line, we have used Hölder’s inequality. Here \({\mathcal {F}}(t) = \int _0^t F(t) \;\textrm{d}t\) is
Notice that by Young’s inequality, for \(t \ge 0\)
In particular, for \(0\le t \le M\)
and for \(t\ge M\)
Hence,
Using Lemma 2.2 for the second term in the left-hand side of (3.16) and (3.17) in the first term, we obtain
Let \(w(x,t) = ((u-v)^{+}_M +\delta )^{\frac{\beta }{p}}\). Since \(\delta \le (u-v)^{+}_M + \delta \), we see that
Using (3.19) in (3.18), we get
By (2.4), we have
Using this in (3.20) and since J has length \(R^{sp}\), we arrive at
Upon multiplying both sides by \(\frac{3 \cdot 2^{p-1} \cdot \beta ^{p-1}}{\delta }\), this implies
Since \(\delta \ge 1\) and \(p \ge 2\), for \(\beta \ge 1\) we have
Using this in (3.21), we get
where \(C=C(n,p)\). Now we consider two cases depending on whether \(sp \ne n\) or \(sp=n\).
Case \(sp \ne n\): Notice that since \(\nu > 0 \), if we take \(\vartheta = 1+ \frac{sp\nu }{n}\), the exponents \((\vartheta r^\prime )^\prime , (\vartheta q^\prime )^\prime \) satisfy the condition of Lemma 2.8. Indeed,
As \(w- \delta ^{\frac{\beta }{p}}\) does vanish in \(B_{R}(x_0)^c\), using Lemma 2.8 for the exponents \((\vartheta q^\prime )^\prime \) and \((\vartheta r^\prime )^\prime \) we get
Here we have used that \(w- \delta ^{\frac{\beta }{p}}\) is nonnegative as well as the fact that \([w]_{W^{s,p}({\mathbb {R}}^n)}\) does not change by subtracting a constant from w. Hence, by (3.19) and (3.23) we obtain
Observe that \((\frac{1}{\vartheta }-1)(sp\nu + n)= - \frac{\vartheta -1}{\vartheta }(sp\nu +n) = -sp\nu \). Furthermore, recalling the definition of \(\delta \) (3.15) whenever \(\delta > 1\) we have
When \(\delta = 1 \), it is straightforward to verify that
Inserting these into (3.24), we arrive at
Now we iterate this inequality with the following choice of exponents
With the notation
(3.25) reads
Iterating this yields
Since \(\vartheta >1 \), we have the following convergent series
and
In the last line, we have used that \(p-1\ge 1\) and \(\delta \ge 1\). Inserting (3.27) to (3.26) and sending m to infinity, we obtain
Since the above estimate is independent of M, we get
which is the desired result.
Case sp=n. Here we use the critical case of Sobolev–Morrey inequality, (2.8) with
This applied for the second term in the left-hand side of (3.22) implies
We replace the constant C(n, s, p, l) with \( \max {\lbrace 1, C(n,s,p,l) \rbrace }\), and multiply both sides with it to arrive at
In the last line, we have used that since \(sp=n\) we have
We have also used the following inequality which we have discussed in the case \(sp\ne n\):
Now we choose \(\vartheta = 2-\frac{1}{r}- \frac{1}{q} - \frac{p}{lr^\prime }\). Notice that due to the choice of l, (3.28), we have \(\vartheta > 1\). Then the exponents \((\vartheta r^\prime )^\prime \) and \((\vartheta q^\prime )^\prime \) satisfy
Therefore, we can apply Lemma 2.9 with the exponents \((\vartheta r^\prime )^\prime \) and \((\vartheta q^\prime )^\prime \) to (3.29) to arrive at
We apply (3.30) with the exponents
Let
Then (3.30) reads
By iterating the above inequality, we get
Since \(\vartheta > 1\), we have the following convergent series
and
By (3.8) in Lemma 3.2, we obtain
Inserting this into (3.30), and sending m to infinity, we get
Hence, we arrive at the desired estimate
\(\square \)
Notice that \(-u\) is a solution to the same type of problem, and we can apply the above proposition to \(-u\). Since \(-v\) is the (s, p)-caloric replacement of \(-u\), we get the same bound on \(\Vert (-u+v)^+\Vert _{L^\infty (Q_{R,R^{sp}})}\); as a result, we get a bound on the \(\Vert u-v\Vert _{L^\infty (Q_{R,R^{sp}})}\).
Corollary 3.5
Let u be a solution of \(\partial _t u + (-\Delta _p)^s u=f\) in \(Q_{\sigma ,\sigma ^{sp}}(x_0,T_0)\) with \(f \in L^{q,r}_{\textrm{loc}} (Q_{\sigma ,\sigma ^{sp}}(x_0,T_0))\) such that
In addition, assume that \(r\ge p^\prime \),
Let v be the (s, p)-caloric replacement of u in \(Q_{R,R^{s p}}(x_0,T_0)\), with \(R< \sigma \).
If \(sp \ne n\), then
where \(\nu = 1-\frac{1}{r} - \frac{n}{spq}\) and \( \vartheta = 1+\frac{sp\nu }{n}. \)
If \(sp=n\), then for any l such that \( \frac{p}{r^\prime }(1-\frac{1}{r} - \frac{1}{q})^{-1}<l < \infty \), we have
where \(\vartheta = 2-\frac{1}{r}- \frac{1}{q} - \frac{p}{lr^\prime }\) and \(\nu = 1-\frac{1}{r} - \frac{1}{q}\).
Now we combine the local boundedness results for the equations with zero right-hand side (see [45] and also [22]) with Proposition 3.4 to prove local boundedness for the equation with nonzero right-hand side.
By [10, Theorem 1.1] with \(q=p\) and \(\sigma = \frac{1}{2}\), we have
where \(\beta =\frac{2\,s+3n-\frac{2n}{p}}{n+s} \) and C depends on n, s and p. By Hölder’s inequality, we have
As \(p\ge 2\), we have \(\frac{2}{p^2}\le \frac{1}{p}\) and \(\frac{\beta }{(\beta -1)p^2}\le \frac{1}{p}\). Hence, we arrive at
Proof of Theorem 1.1
For u, a local weak solution of
we consider v to be the (s, p)-caloric replacement in \(Q_{R,R^{sp}}(x_0,T_0)\),
As mentioned in the proof of Proposition 3.5, our assumptions ensure that we can use Proposition 3.1 and v exists. Using (3.32), we arrive at
Using Lemma 2.3 in (3.33), we arrive at
where \(C=C(n,s,p)\). Finally, using Proposition 3.4 to estimate the term \(\Vert u-v\Vert _{L^\infty (Q_{R,R^{sp}})}\), in (3.34) we get the desired result. Here the estimate is written in the case \(sp\ne n\)
\(\square \)
Theorem 3.6
Let \(f \in L^{q,r}(Q_{R_1,R_1^{sp}}(z,T_1))\) with
In addition, assume that \(r\ge p^\prime \),
If u is a weak solution of the equation
such that
then u is locally Hölder continuous in time and space. In particular, there exists a \(\zeta >0\), such that for \(\sigma <1\), \((x_1,t_1), \; (x_2,t_2) \in Q_{\sigma R_1,(\sigma R_1)^{sp}}(z,T_1) \), there holds
with C depending on n, s, p and \(\sigma \), and
Proof
Take a cylinder \(Q_{\sigma R_1,(\sigma R_1)^{sp}}(z,T_1) \subset Q_{R_1,R_1^{sp}}(z,T_1)\) and let \(d:=\min { \lbrace R_1(1-\sigma ),R_1(1-\sigma ^{sp})^{\frac{1}{sp}}\rbrace } > 0 \). For any point, \((x_0,T_0) \in Q_{\sigma R_1,(\sigma R_1)^{sp}}(z,T_1) \) consider the (s, p)-caloric replacement of u in the cylinder \(Q_{R,R^{sp}}(x_0,T_0)\) with \(R \le \min { \lbrace 1, d \rbrace }\). The choice of d implies that \(Q_{R,R^{sp}}(x_0,T_0)\subset Q_{ R_1,R_1^{sp}}(z,t) \). First, we observe that:
For \(\rho \le \frac{R}{2}\), v is Hölder continuous in \(Q_{\rho ,\rho ^{sp}}(x_0,T_0)\) by Theorem 5.1, and by the mean value theorem, there is a point \(({\tilde{x}}_0,{\tilde{t}}_0)\in Q_{\rho ,\rho ^{sp}}\) such that \({\bar{v}}_{x_0,t_0}= v({\tilde{x}}_0,{\tilde{t}}_0)\). With the notation
Theorem 5.1 implies:
with \(C=C(n,s,p)\). Therefore,
where the constants C depends on n, s and p, and we have defined \(\delta := \min { \bigl \lbrace \frac{\Theta }{2}, \frac{\Gamma }{2}\bigr \rbrace }\).
Moreover, by Lemma 3.2
where \(\xi \) is defined in Lemma 3.2. Notice that \(\xi >0\) by our assumptions on q and r. Inserting (3.37) and (3.36) in (3.35), we arrive at
Using Corollary 3.5, we get:
with \(\vartheta \) and \(\nu \) defined in Corollary 3.5; here, the estimate is only written in the case \(sp \ne n\) for simplicity. Since \(Q_{R,R^{sp}}(x_0,T_0)\subset Q_{ R_1,R_1^{sp}}(z,T_1)\), the above expression is less than
Concerning the tail term, since \(B_{R}(x_0) \subset B_{R_1}(z)\), using Lemma 2.4 we have
and by the choice of the radii, we have
Hence, taking the supremum in time and using Minkowski’s inequality in (3.38), we arrive at
where the above constant C depends on \(n, s \text {and }p\). In conclusion,
Now we make the choice \(\rho =\frac{R^\theta }{2}\) with
This yields
for any \(0< \rho < \frac{\min {\lbrace 1,d \rbrace }^\theta }{2}\), where
For values of \(\rho \ge \frac{\min {\lbrace 1,d \rbrace }^\theta }{2}\),
We can then conclude that for any cylinder of arbitrary size we have
with C depending on
In particular, one can obtain
Now we use the characterization of the Campanato spaces in \({\mathbb {R}}^{n+1}\) with a general metric in [30], see also [20]. Our setting does not fit directly in the context considered there, since we only work with cylinders that are one-sided in the time direction that is \((t-r^{sp},t]\times B_r(x)\) instead of \((t-r^{sp},t+r^{sp})\times B_r(x)\). Still, if you follow the proof in [30] with small modifications, you can also conclude the result in this setting.
In the case of \(sp \ge 1\), using [30, Theorem 3.2] we get the Hölder continuity of u with exponent \(\zeta \) in \(Q_{\sigma R, (\sigma R)^{sp}}\) with respect to the metric
for which the balls of radius r are of the form \((t-r^{sp},t+r^{sp})\times B_r(x)\), which means
Here C depends on n, s, p and \(\sigma \). In the case of \(sp< 1\), we use the metric
The balls of radius r are of the form \((t-r,t+r)\times B_{r^{\frac{1}{sp}}}(x)\). Hence, we have a decay of order \(r^{\frac{\xi }{sp} p}\) of the average of u on the half balls. [30, Theorem 3.2] implies the following Hölder continuity on \(Q_{\sigma R_1,(\sigma R_1)^{sp}}\)
\(\square \)
Lemma 3.7
(Stability in \(L^\infty \)) Let \(f \in L^{q,r}_{\textrm{loc}}(Q_{2R,(2R)^{sp}})\) with
In addition, assume that \(r\ge p^\prime \),
Let u be a local weak solution to the equation
with
and
Consider the (s, p)-caloric replacement
Then for \(\sigma < 1\), there is a \(\delta _{M,R,\sigma }(\omega )\) such that
and \(\delta _{M,R, \sigma }(\omega )\) converges to 0 as \(\omega \) goes to 0.
Proof
The existence of such a bound follows immediately from Corollary 3.5.
To show the convergence of \(\sigma _{M,R,\sigma }\) to zero, we argue by contradiction, suppose that there is a sequence \(f_n \in L^{q,r}(Q_{R,R^{s \, p}})\) and \(u_n\) such that
but
Using (3.12) from Lemma 3.2, we have
By assumption, \(u_n\) is uniformly bounded in \(L^\infty (Q_{ R, R^{sp}})\). Now we show that \(\varphi _n\) is also uniformly bounded in \(L^\infty (Q_{R, R^{sp}})\).
By Corollary 3.5,
Since \(\Vert f_n\Vert _{L^{q,r}(Q_{R,R^{s \, p}})}^{p^\prime }\) is uniformly bounded, (3.43) and (3.42) give us a uniform bound on \( \Vert \varphi _n\Vert _{L^\infty (Q_{ R, R^{s\,p}})} \).
Now we are in a position to use Theorem 3.6 for both of the sequences \(u_n\) and \(\varphi _n\), which gives us a uniform bound on the Hölder seminorms of \(u_n\) and \(\varphi _n\) in \(Q_{\sigma R,(\sigma R)^{s p}}\). Therefore, by Arzela–Ascoli’s theorem \(u_n - \varphi _n\) has a uniformly convergent subsequence in \(Q_{\sigma R,(\sigma R)^{sp}}\). By (3.41), the limit is 0, contradicting (3.40). \(\square \)
4 Improved Hölder regularity for nonhomogeneous equation
Proposition 4.1
Let \(f\in L^{q,r}(Q_{1,2})\) with q, r satisfying
In addition, assume that \(r\ge p^\prime \),
Let u be a weak solution of \(u_t + (-\Delta _p)^s u=f\) in \(Q_{1,2}\) that satisfies
Then there exists \(\omega \) such that if
u is locally Hölder continuous in \(Q_{\frac{1}{2},\frac{1}{2^{sp}}}\) with exponents \(\alpha \) in space and \(\frac{\alpha }{sp-(p-2)\alpha }\) in time, as long as
Recall that \(\Theta = \min { \left\{ \frac{sp}{p-1},1 \right\} }\).
More precisely, for \((x_1,t_1),\; (x_2,t_2) \in Q_{\frac{1}{2},\frac{1}{2^{sp}}}\) we have
Proof
Step 1: Decay at the origin.
For this part, we prove a decay at the origin for u under the assumptions
Here \(\omega >0\) is a small number to be determined later which depends on n, s, p and \(\alpha \). We introduce the parabolic cylinder
with \(\beta =sp-(p-2)\alpha \). We show that for any exponent \(\alpha \) satisfying (4.1), the following holds for \(r< 1\)
It is enough to prove the inequality for a sequence of radii \( (r_k)_{k=0}^\infty ,\; r_k= \lambda ^k \), for some \(\lambda < 1\). Without loss of generality, we assume \(u(0,0)=0\). Consider the rescaled functions
with \(\lambda \) small enough to be determined later. We will prove the following by induction,
For \(k=0\), (4.3) follows from our assumptions (4.2).
Observe that
With \(\beta = sp - (p-2)\alpha \), \(v_k(x,t)\) solves
Moreover,
Since \(\lambda <1\), and the exponent of \(\lambda \) is nonnegative by (4.1), we get \(\Vert f_{k}\Vert _{L^{q,r}(G_1)} \le \omega \).
Assume that (4.3) holds for k. Now we prove that it holds for \(k+1\). Consider the (s, p)-caloric replacement of \(v_k(x,t)\) in \(Q_{1,1}\), say \(\varphi _k(x,t)\). Then
By Theorem 5.1, \(\varphi _k\) is locally Hölder continuous in \(Q_{1,1}\), and for \((x,t) \in Q_{\frac{1}{2}, \frac{1}{2^{sp}}}\),
Here we take \(\varepsilon = \frac{\Theta -\alpha }{2}\). Since \(\Vert f_k\Vert _{L^{q,r}(Q_{1,1})} \le \omega \), Lemma 3.7 implies
In Theorem 5.1, the Hölder constants are bounded by
Therefore, by (4.3) we have
By Corollary 3.5,
This is a bound independent of k. We can take \(\omega \) to be less than 1 and take \(C_1=C(n,s,p)( 3 + 2C(n,s,p,q,r),\) with the C(n, s, p, q, r) coming from Corollary 3.5, so that the constants \(C_1,\; C_2\) are independent of \(\omega \) as well.
Now we proceed and prove (4.3) for \(k+1\). First, we state our choice of \(\lambda \)
Since \(\lambda < \frac{1}{4}\), and \(\lambda ^\beta < \frac{1}{4^{s\, p}}\), \(Q_{\lambda , \lambda ^{\beta }} \subset Q_{\frac{1}{4},\frac{1}{4^{s \, p}}}\). Therefore, from (4.4) we obtain
Notice that \(\beta \Gamma \ge \Theta \), by the above choice of \(\beta \). Thus,
Recall that \(\varepsilon = \frac{\Theta -\alpha }{2}\) and by the assumption (4.5)
Now we choose \(\omega \) so that
This is possible since \(\delta (\omega )\) converges to zero as \(\omega \rightarrow 0\). Then, (4.6) implies
which translates to
which is the first part of (4.3). For the second part, we want to show
We split the integral into three parts. Using the induction hypothesis,
Moreover, \(\Vert v_k\Vert _{L^\infty (G_1)} \le 1\), and hence,
For remaining part, we transfer the estimate (4.4) to \(v_{k+1}\) and obtain
In particular, since \(\lambda ^\beta \le \frac{1}{4^{sp}}\), \(Q_{\frac{1}{4\lambda },1} \subset Q_{\frac{1}{4\lambda }, \frac{1}{4^{sp}\lambda ^\beta }}\), and \(\delta (\omega ) \le \lambda ^{\Theta } \le \lambda ^{\Theta -\varepsilon }\) we get
Therefore,
Hence,
Using the assumption (4.5) on \(\lambda \), we obtain
Step 2: Regularity in a cylinder. We choose \(\alpha \) as in (4.1) and let \(\omega \) be as in Step 1. For a point \((x_0,t_0) \in Q_{\frac{1}{2}, \frac{1}{2^{sp}}}\), define
where \(L = 2^\frac{n}{p-1}(1+|B_1|)^{\frac{1}{p-1}}\). Then \({\tilde{u}}\) is a solution of
By the choice of L, \({\tilde{u}}\) satisfies the conditions (4.2) in Step 1. Since \(L\ge 1\), we immediately have
since \(Q_{\frac{1}{2},\frac{L^{2-p}}{2^{sp}}}(x_0,t_0) \subset Q_{1,2}\). As for the \(L^{q,r}\) norm of \({\tilde{f}}\), we have
Here we have used \(1-\frac{1}{r}-\frac{n}{spq} > 0\). Notice that in the case of \(sp \ge n\), we are assuming \(1-\frac{1}{r} -\frac{1}{q} > 0\) which is a stronger assumption. Now we verify the assumption on the tail.
Now we can apply Step 1 to \({\tilde{u}}\) and we get the decay
or in other words
In terms of u, this means
Now take two points \((x_1,t_1)\,, \, (x_2,t_2) \in Q_{\frac{1}{2}, \frac{1}{2^{sp}}}\) and split the line joining them into \(1+ [L^{p-2}]\) pieces, say \((y_i,\tau _i)_{i=0}^{1+[L^{p-2}]}\) with \((x_1,t_1)=(y_0,\tau _0)\), \((x_2,t_2)=(y_{1+[L^{p-2}]},\tau _{1+[L^{p-2}]})\), \(|y_{i+1}-y_i|= \frac{|x_2-x_1|}{1+[L^{p-2}]} < \frac{1}{2}\) and \(|\tau _{i+1}- \tau _i|= \frac{|t_2-t_1|}{1+[L^{p-2}]} < \frac{1}{2^{sp}L^{p-2}}\) so that \((y_{i+1},\tau _{i+1}) \in Q_{\frac{1}{2}, \frac{1}{2^{sp}L^{p-2}}}(y_i,\tau _i)\). By (4.8) applied in each of \(Q_{\frac{1}{2}, \frac{1}{2^{sp}L^{p-2}}}(y_i,\tau _i)\) obtain
\(\square \)
Now we prove the Hölder regularity at any scale.
Proof of Theorem 1.2
We will consider the rescaled functions
with
where \(\omega = \omega (n,s,p,q,r,\alpha )\) is the same as in the proof of Proposition 4.1 and \(\iota \in [ -(R/2)^{sp} (1-\mu ^{2-p}),0]\). The interval \([ -(R/2)^{sp} (1-\mu ^{2-p}),0]\) is chosen so that the cylinders \(Q_{\frac{R}{2},\frac{\mu ^{2-p} R^{sp}}{2^{sp}}}(x_0,T_0+\iota )\) cover all of \(Q_{\frac{R}{2},(\frac{R}{2})^{sp}}(x_0,T_0)\) by varying \(\iota \) over. Note that for these choices of \(\iota \) we have \(Q_{R,2\mu ^{2-p}R^{sp}}(x_0,T_0+\iota )\subset Q_{R,2R^{sp}}(x_0,T_0)\). Then \({\tilde{u}}\) is a solution of
We now verify that \({\tilde{u}}_\iota \) satisfies the conditions of Proposition 4.1. The \(L^{q,r}\) norm of the right-hand side is
The \(L^\infty \) norm of \({\tilde{u}}_\iota \) satisfies
Similarly
Hence, using Proposition 4.1 for \({\tilde{u}}_\iota \), we get
with \(C= C(n,s,p,q,r,\alpha ) \). This translates to
for \((x_1,\tau _1), \; (x_2,\tau _2) \in Q_{\frac{R}{2}, \frac{R^{s \, p}\mu ^{2-p}}{2^{s \, p}}}(x_0,T_0+\iota )\). Now we vary \(\iota \) to obtain an estimate in the whole \(Q_{\frac{R}{2}, (\frac{R}{2})^{s \, p}}\). Specifically we split the interval \([t_1,t_2]\) into \(1+ \lfloor \mu ^{p-2} \rfloor \) pieces, say \([\tau _{i+1},\tau _i]\), with \(\tau _i - \tau _{i+1}=\frac{|t_2-t_1|}{1+\lfloor \mu ^{p-2} \rfloor }\), \(\tau _0=t_2\), and \(\tau _{\lfloor 1+\mu ^{p-2} \rfloor } = t_1\). Using (4.9), we obtain
which concludes the desired result. \(\square \)
References
B. Abdellaoui, A. Attar, R. Bentifour, I. Peral, On fractional\(p\)-Laplacian parabolic problem with general data, Ann. Mat. Pura Appl., 197 (2018), 329–356.
K. Adimurthi, H. Prasad, V. Tewary, Local Hölder regularity for nonlocal parabolic p-Laplace equations, arXiv:2205.09695, (2022), 1–31.
F. Anceschi, M. Piccinini, Boundedness estimates for nonlinear nonlocal kinetic Kolmogorov-Fokker-Planck equations, arXiv:2301.06334, (2023), 1–26.
F. Andreu-Vaillo, J. Mazón, J.D. Rossi, J.J. Toledo-Melero, Nonlocal diffusion problems Mathematical Surveys and Monographs, 165. American Mathematical Society, Providence, RI; Real Sociedad Matematica Espanñla, Madrid, (2010).
D.G. Aronson, J. Serrin, Local behavior of solutions of quasilinear parabolic equations, Arch. Rational Mech. Anal. 25 (1967), 81–122.
L. Brasco, E. Lindgren, A. Schikorra, Higher Hölder regularity for the fractional\(p\)-Laplacian in the superquadratic case,Adv. Math. 338 (2018), 782–846.
L. Brasco - E. Lindgren - M. Strömqvist Continuity of solutions to a nonlinear fractional diffusion equation J. Evol. Equ. 21 (2021), no. 4, 4319–4381.
L. Brasco, E. Parini, The second eigenvalue of the fractional\(p-\)Laplacian, Adv. Calc. Var. 9 (2016), no. 4, 323–355.
L. Brasco, F. Prinari, and A.C. Zagati, Sobolev embeddings and distance functionsarXiv:2301.13026, (2023), 1–42, to appear in Adv. Calc. Var.
S. Byun and K. Kim. A Hölder estimate with an optimal tail for nonlocal parabolic p-Laplace equations, to appear in Annali di Matematica (2023) https://doi.org/10.1007/s10231-023-01355-6.
S. Byun, H. Kim, and K. Kim. Higher Hölder regularity for nonlocal parabolic equations with irregular kernels, J. Evol. Equ. 23 (2023), no. 3, Paper No. 53, 1–59.
S. Byun, K. Kim, and K. Kumar. Calderon-Zygmund theory of nonlocal parabolic equations with discontinuous coefficients, J. Differential Equations 371 (2023), 231–259.
L.A. Caffarelli, C. Chan, and A. Vasseur, Regularity theory for parabolic nonlinear integral operators, J.Amer.Math.Soc., 23 (2011), no. 3, 849–869.
L. Caffarelli, L. Silvestre Regularity results for nonlocal equations by approximation, Arch. Ration. Mech. Anal., 200 (2011), no. 1, 59–88.
L. Caffarelli, L. SilvestreThe Evans-Krylov theorem for nonlocal fully nonlinear equations, Ann. of Math. (2), 174 (2011), no. 2, 1163–1187.
L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasigeostrophic equation, Ann. of Math. (2), 171 (2010), 1903–1930.
H. Chang-Lara, G. Dávila, Regularity for solutions of non local parabolic equations, Calc. Var. Partial Differential Equations, 49 (2014), 139–172.
H. Chang-Lara, G. Dávila, Regularity for solutions of non local parabolic equations II, J. Differential Equations, 256 (2014), 130–156.
M. Cozzy, Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: a unified approach via fractional De Giorgi classes, J. Funct. Anal., 272 (11), (2017), 4762–4837.
G. Da Prato, Spazi \(L^{p,\theta }(\Omega ,\; \delta )\)e loro proprietà, Ann. Mat. Pura Appl. (4), 69 (1965), 383–392.
E. DiBenedetto, Degenerate parabolic equations. Universitext. Springer-Verlag, New York, (1993)
M. Ding, C. Zhang, and S. Zhou, Local boundedness and Hölder continuity for the parabolic fractional p-Laplace equations, Calc. Var. 60, 38 (2021)
A. Di Castro, T. Kuusi, and G. Palatucci. Nonlocal Harnack inequalities. J. Funct, Anal., 267 (6), (2014), 1807–1836.
A. Di Castro, T. Kuusi, and G. Palatucci. Local behavior of fractional p-minimizers, Ann. Inst. H. Poincare Anal. Non Lineaire, 33 (5) (2016), 1279–1299.
L. Diening and S. Nowak. Calderón-Zygmund estimates for the fractional p-Laplacian, arXiv:2303.02116, (2023).
M. Fall. Regularity results for nonlocal equations and applications, Calc. Var. Partial Differential Equations, 59 (5) (2020) No. 181, 53.
M. Felsinger and M. Kassmann. Local regularity for parabolic nonlocal operators, Comm. Partial Differential Equations, 38 (9),(2013) 1539–1573.
X. Fernandez-Real and X. Ros-Oton. Regularity theory for general stable operators: parabolic equations. J. Funct. Anal., 272, (2017), no 10, 4165–4221.
J. Giacomoni, D. Kumar, K. Sreenadh Hölder regularity results for parabolic nonlocal double phase problems, arxiv:2112.04287v3
P. Gorka, Campanato theorem on metric measure spaces, Ann. Acad. Sci. Fenn. Math., 34 (2009), 523–528
H. Ishii, G. Nakamura. A class of integral equations and approximation of\(p\)-Laplace equations, Calc. Var. Partial Differential Equations 37, no. 3-4, 485–522. (2010)
M. Kassmann. A priori estimates for integro-differential operators with measurable kernels, Calc. Var. Partial Differential Equations, 34 (2009), no. 1, 1–21.
M. Kassmann and M. Weidner. Nonlocal operators related to nonsymmetric forms II: Harnack inequalities, arXiv:2205.05531, (2022), 1–56.
M. Kassmann and M. Weidner. The parabolic Harnack inequality for nonlocal equations, arXiv:2303.05975, (2023), 1–27.
T. Kuusi, G.Mingione, and Y. Sire. Nonlocal equations with measure data, Comm. Math. Phys. 337 (2015), no.3, 1317–1368.
T. Kuusi, G. Mingione. Guide to nonlinear potential estimates, Bull. Math. Sci. 4 (2014), no.1, 1–82.
N. Liao. Hölder regularity for parabolic fractional\(p\)-Laplacian, (2022), arXiv:2205.10111 1–26.
V. Liskevich, I. Skrypnik, and Z. Sobol. Estimates of solutions for the parabolic p-Laplacian equation with measure via parabolic nonlinear potentials, Commun. Pure Appl. Anal. 12 (2013), no. 4, 1731–1744.
J.M. Mazón, J.D. Rossi and J. Toledo, Fractional\(p\)-Laplacian evolution equations, J. Math. Pures Appl. (9), 105 (2016), no. 6, 810–844.
S. Nowak, Higher Holder regularity for nonlocal equations with irregular kernel, Calc. Var. Partial Differential Equations, 60 (2021) no. 1, Paper No. 24, 1–37.
S. Nowak. Regularity theory for nonlocal equations with VMO coefficients, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 40 (2023), no.1, 61–132.
Dimitri Puhst, On the evolutianary fractional p-Laplacian, Appl. Math. Res. Express. AMRX (2015), no. 2, 253–273.
X. Ros-Oton and J. Serra. Regularity theory for general stable operators, J. Differential Equations 260 (2016), no. 12, 8675–8715
R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs, 49, American Mathematical Society, Providence, RI, (1997).
M. Strömqvist, Local boundedness of solutions to non-local parabolic equations modeled on the fractional p-Laplacian, Journal of Differential Equations 266 (2019), 7948–7979.
M. Strömqvist, Harnack’s inequality for parabolic nonlocal equations, Ann. Inst. H. Poincare C Anal. Non Lineaire 36 (2019), no. 6, 1709–1745.
E. Teixeira, J. Urbano.A geometric tangential approach to sharp regularity for degenerate evolution equations, Anal. PDE 7, (2014), no. 3, 733–744.
J. L. Vázquez, The Dirichlet problem for the fractional\(p\)-Laplacian evolution equation, J. Differential Equations, 260 (2016), 6038–6056.
J. L. Vázquez, The evolution fractional\(p\)-Laplacian equation in\({{\mathbb{R}}}^N\). Fundamental solution and asymptotic behaviour, Nonlinear Anal., 199 (2020), 112034, 1–32.
V. Vespri, \(L^\infty \)-estimates for nonlinear parabolic equations with natural growth conditions. Rend. Sem. Mat. Univ. Padova 90 (1993), 1–8.
M. Warma, Local Lipschitz continuity of the inverse of the fractional\(p\)-Laplacian, Hölder type continuity and continuous dependence of solutions to associated parabolic equations on bounded domains, Nonlinear Anal., 135 (2016), 129–157.
Acknowledgements
The author warmly thanks Erik Lindgren for introducing the problem, proofreading this paper, for his helpful comments, and for long hours of fruitful discussions. During the development of this paper, I have been a Ph.D. student at Uppsala University. In particular, I wish to express my gratitude to the Department of Mathematics at Uppsala University for its warm and hospitable research environment. The author also wishes to thank the anonymous referee for their careful reading and helpful comments which have resulted in improving the quality of this article.
Funding
Open access funding provided by Royal Institute of Technology. The author has partially been supported by the Swedish Research Council, Grant No. 2017-03736.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
This paper was finalized while I was participating in the program geometric aspects of nonlinear partial differential equations at Mittag-Leffler Institute in Djursholm, Sweden, during the fall of 2022. The research program is supported by Swedish Research Council grant no. 2016-06596
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendix A
In this section, we spell out the necessary modifications to prove the following theorem 5.1 which is a modified version of [7, Theorem 1.2]. As it is explained in [7, Remark 1.4] one can obtain the conclusions of [7, Theorem 1.2] under the weaker assumptions \( u\in L^{\infty }_{\textrm{loc}}(I;L^\infty _{\textrm{loc}}(\Omega )) \cap L^\infty _{\textrm{loc}} (I; L_{sp}^{p-1}({\mathbb {R}}^n))\), instead of \(u \in L^{\infty }_{\textrm{loc}}(I;L^\infty ({\mathbb {R}}^n)) \).
Theorem 5.1
Let \(\Omega \subset {\mathbb {R}}^n\) be a bounded and open set, \(I=(t_0,t_1]\), \(p\ge 2\) and \(0<s<1\). Suppose u is a local weak solution of
such that
Define the exponents
Then
More precisely, for every \(0<\delta <\Theta (s,p)\), \(0<\gamma <\Gamma (s,p)\), \(R>0\), \(x_0\in \Omega \) and \(T_0\) such that
there exists a constant \(C=C(n,s,p,\delta , \gamma ,\sigma )>0\) such that
for any \((x_1,\tau _1),\,(x_2,\tau _2)\in Q_{\sigma R,(\sigma R)^{s\,p}}(x_0,T_0)\).
First we reproduce a modified version of [7, Proposition 4.1], where instead of a global \(L^\infty \) bound we assume \(\Vert u\Vert _{L^\infty (B_1 \times [-1,0])} +\sup _{t\in [-1,0]} \textrm{Tail}_{p-1,sp}(u;0,1)) \le 1\). Before stating the proposition, let us recall the following notations from [7]:
and
Proposition 5.2
Assume \(p\ge 2\) and \(0<s<1\). Let u be a local weak solution of \(u_t+(-\Delta _p)^s u=0\) in \(B_2\times (-2,0]\). We assume that
and that, for some \(q\ge p\) and \(0<h_0<1/10\), we have
for a radius \(4\,h_0<R\le 1-5\,h_0\) and two time instants \(-1<T_0<T_1\le 0\). Then we have
for every \(0<\mu <T_1-T_0\). Here \(C=C(n,s,p,q,h_0,\mu )>0\) and \(C\nearrow +\infty \) as \(h_0\searrow 0\) or \(\mu \searrow 0\).
Proof
In the proof of [7, Proposition 4.1], the \(L^\infty ({\mathbb {R}}^n \times [0,1])\) boundedness is only used in Step 3, in the estimation of the nonlocal terms \({\mathcal {I}}_2\) and \({\mathcal {I}}_3\), which are defined by
and
We also recall the definition of \({\tilde{{\mathcal {I}}}}_2\) and \({\tilde{{\mathcal {I}}}}_3\)
where \(\tau \) is smooth function \(0\le \tau \le 1\) such that
The general argument is the same, but instead of using the \(L^\infty \) norm of u(y) we can keep the inequality as it is and write
where \(x \in B_{R-2h_0} \) and \(4h_0< R < 1-5h_0\). Therefore, \(|x-y| \ge (1-\frac{R-2h_0}{R})|y| \ge C(h_0) |y|\) and we get
Now
and for \(u_h\)
Here we have used \(B_R(h) \subset B_1\), and \(\frac{|y-h|}{|y|} = |\frac{y}{|y|}-\frac{h}{|y|}| \ge |\frac{y}{|y|}| - |\frac{h}{|y|}| \ge 1- |\frac{h_0}{R-h_0}| \ge \frac{2}{3}\). Using this, we get
and we can conclude
which is the same as equation (4.6) in [7]. \(\square \)
We can estimate the \(W^{s,p}\) seminorm of a solution as follows. The proof follows the argument in [7, Lemma 7.1].
Lemma 5.3
Let \(p\ge 2\) and \(0<s<1\). Let u be a local weak solution of
such that \(u\in L^\infty (B_{2R}\times [-R^{s\,p},0])\). Then
for some \(C=C(n,s,p)>0\).
Proof
Without loss of generality, we may suppose that \(x_0=0\). Let
Then \({{\widetilde{u}}}\) is a local weak solution in \(B_2\times (-2\,R^{s\,p},0]\) and \({{\widetilde{u}}}\ge 1\) in \(B_{2R}\times [-R^{s\,p},0]\). We choose \(\varphi \) and \(\psi \) exactly as in [7, Lemma 7.1], that is,
and
Then for \(\varphi (x,t)= \eta (x) \varphi (t)\), we get
The only difference in the proof is in estimating the term
Noticing that for \(x \in {{\,\textrm{supp}\,}}\eta \subset B_{\frac{3}{2}R}\) we have \(\frac{|x-y|}{|y|} \ge 1- \frac{|x|}{|y|} \ge 1-\frac{3/2 R}{2R} = \frac{1}{4}\), we get
\(\square \)
We can now prove the following modified version of [7, Theorem 4.2].
Theorem 5.4
(Spatial almost \(C^s\) regularity) Let \(\Omega \subset {\mathbb {R}}^n\) be a bounded and open set, \(I=(t_0,t_1]\), \(p\ge 2\) and \(0<s<1\). Suppose u is a local weak solution of
such that \(u\in L^\infty _{\textrm{loc}}(I;L^\infty (\Omega )) \cap L^\infty _{\textrm{loc}}(I;L_{sp}^{p-1}({\mathbb {R}}^n))\). Then \(u\in C_{x,\textrm{loc}}^\delta (\Omega \times I)\) for every \(0<\delta <s\).
More precisely, for every \(0<\delta <s\), \(R>0\) and every \((x_0,T_0)\) such that
there exists a constant \(C=C(n,s,p,\delta )>0\) such that
Proof
The proof is essentially the same as the proof of [7, Theorem 4.2]. We assume for simplicity that \(x_0=0\) and \(T_0=0\), and set
Notice that by Lemma 5.3 we have
Let \(\alpha \in [-R^{s\,p}(1-{\mathcal {M}}_R^{2-p}),0]\) and define
Then \(u_{R,\alpha }(x,t)\) is a local weak solution of
that satisfies
This function satisfies the assumption of Proposition 5.2, and we can do the same argument as in [7] to obtain
for a C independent of \(\alpha \) and by scaling back we get
By varying \(\alpha \in [-R^{s\,p}(1-{\mathcal {M}}_R^{2-p}),0]\), we get the desired result. \(\square \)
We now address the improved regularity and start with the following modified version of [7, Proposition 5.1].
Proposition 5.5
Assume \(p\ge 2\) and \(0<s<1\). Let u be a local weak solution of \(u_t+(-\Delta _p)^s u=0\) in \(B_2\times (-2,0]\), such that
Assume further that for some \(0<h_0<1/10\) and \(\vartheta <1\), \(\beta \ge 2\) such that \((1+\vartheta \, \beta )/\beta <1\), we have
for a radius \(4\,h_0<R\le 1-5\,h_0\) and two time instants \(-1<T_0< T_1\le 0\). Then
for every \(0<\mu <T_1-T_0\). Here C depends on the n, \(h_0\), s, p, \(\mu \) and \(\beta \).
Proof
The only major difference from the proof of Proposition 5.2 is in the estimation of term \({\mathcal {I}}_{1 1}\) and it can be treated in the exact same way as in the proof of [7, Proposition 5.1]. \(\square \)
Using the previous proposition with the same type of modifications as in the proof of Theorem 5.4, we can state the following version of [7, Theorem 5.2].
Theorem 5.6
Let \(\Omega \) be a bounded and open set, let \(I=(t_0,t_1]\), \(p\ge 2\) and \(0<s<1\). Suppose u is a local weak solution of
such that \(u\in L^\infty _{\textrm{loc}}(I;L^\infty _{\textrm{loc}}(\Omega )) \cap L^\infty _{\textrm{loc}} (I; L_{sp}^{p-1}({\mathbb {R}}^n))\). Then \(u\in C^\delta _{x,\textrm{loc}}(\Omega \times I)\) for every \(0<\delta <\Theta (s,p)\), where \(\Theta (s,p)\) is defined in (5.2).
More precisely, for every \(0<\delta <\Theta (s,p)\), \(R>0\), \(x_0\in \Omega \) and \(T_0\) such that
there exists a constant \(C=C(n,s,p,\delta )>0\) such that
Now we modify the argument regarding the regularity in time (see [7, Proposition 6.2]).
Proposition 5.7
Suppose that u is a local weak solution of
such that
and
where \(\Theta (s,p)\) is the exponent defined in (5.2). Then there is a constant \(C= C(n,s,p,K_\delta ,\delta )>0\) such that
where
In particular, \(u\in C^\gamma _{t}(Q_{\frac{1}{4},\frac{1}{4}})\) for any \(\gamma <\Gamma (s,p)\), where \(\Gamma (s,p)\) is the exponent defined in (5.2).
Proof
The only part that needs to be modified is the estimation of the nonlocal term \(J_2\)
here \(T_0,T_1 \in (t_0-\theta , t_0)\) with \(T_0<T_1\). We recall that \(0< \theta < \frac{1}{8}\), \(x_0 \in B_{\frac{1}{4}}\), and \(r < \frac{1}{8}\). Thus, \(x\in B_{\frac{r}{2}}(x_0)\) implies \(x\in B_{\frac{5}{16}}\).
For \(y \in B_{\frac{1}{2}}(0)\), assumption (5.8) implies
For \(y \in B_2(0)\setminus B_{\frac{1}{2}}(0)\), the \(L^\infty \) bound on u implies
Also notice that for \(x\in B_{r/2}(x_0)\) and \(y \in {\mathbb {R}}^n {\setminus } B_{r}(x_0)\), we have \(|x-y| \ge \frac{1}{2}|y-x_0|\). Using these, we obtain
\(\square \)
Finally, we are ready to prove a modified version of [7, Theorem 1.1], which is Theorem 5.1.
Proof of Theorem 5.1
Consider a cylinder \(Q_{2\rho ,2\rho ^{sp}}({\tilde{x}},\tau ) \Subset \Omega \times I\), first, we prove the following type of bound on the Hölder seminorm in \(Q_{\rho /4,\rho ^{s\,p}/4}({\tilde{x}},\tau )\), and later with the aid of a covering argument, we conclude the claim of the theorem.
Claim: For any \((x_1,\tau _1),\,(x_2,\tau _2)\in Q_{\rho /4,\rho ^{s\,p}/4}({\tilde{x}},\tau )\) we have
The regularity in space variable has been proved in Theorem 5.6. To prove the part on time regularity, we set
and consider the rescaled functions
for \(\iota \in (-\frac{\rho ^{sp}}{4}(1- {\mathcal {M}}_\rho ^{2-p}),0)\). Then \({\tilde{u}}_{\rho ,\iota }(x,t)\) is a solution of
Moreover, \({\tilde{u}}_{\rho ,\iota }(x,t)\) satisfies the conditions of Proposition 5.7. Indeed by construction
and the estimate (5.8) follows from (5.7) in Theorem 5.6. From Proposition 5.7, we obtain
with \(C= C(n,s,p,\gamma )\) for every \(0< \gamma < \Gamma (s,p)\). By scaling back, this translates to
By varying \(\iota \) with an argument similar to the proof of Theorem 1.2, we arrive at the claim (5.9). We have to point out that the Hölder constant does change, unlike what is suggested in the proof of [7, Theorem 1.1]. Here is a detailed computation
We split the time interval \([t_1,t_2]\) into \(1+ \lfloor {\mathcal {M}}_{\rho }({\tilde{x}},\tau )^{p-2} \rfloor \) pieces, say \([\tau _{i+1},\tau _i]\), with \(\tau _i - \tau _{i+1}=\frac{|t_2-t_1|}{1+\lfloor {\mathcal {M}}_{\rho }({\tilde{x}},\tau )^{p-2} \rfloor }\), \(\tau _0=t_2\), and \(\tau _{\lfloor 1+\mu ^{p-2} \rfloor } = t_1\). Then using (5.10) and the triangle inequality, we get
Now use (5.9) in cylinders of the form
where the radius \(r= \frac{R}{C(n,s,p,\sigma )}\) is so small, such that
Consider a sequence of points \(({\tilde{x}}_i,{\tilde{\tau }}_i)\) on the segment joining \((x_1,\tau _1)\) and \((x_2,\tau _2)\) such that
Using (5.9) together with the triangle inequality, we obtain
with \(C=C(n,s,p,\delta ,\gamma ,\sigma )\), which is the desired result. \(\square \)
Appendix B
Here we will justify the insertion of \(u-v\) and \(|u-v|^{p-2}(u-v)\) as test functions.
Proposition 6.1
Let \(B= B_R(x_0)\) be a ball of radius r, \(B_2=B_{\sigma R}(x_0)\) with \(\sigma > 1\), and \(I=(\tau _0,\tau _1]\) be an interval. Let \(f \in L^{(p_s^\star )^\prime ,p^\prime }(B\times I)\) and assume that \(u \in L^p(I, W^{s,p}(B_2)) \cap L^{p-1}(I;L_{sp}^{p-1}({\mathbb {R}}^n)) \cap C(I;L^2(B))\) is a local weak solution of
with
(in particular, this will be the case under the stronger assumption \(\sup _{t \in I} \textrm{Tail}_{p,sp}(u(\,\bullet \,,t);x_0,R) < \infty \) that we use in this article.) Let \([T_0,T_1] \Subset I\) and let \(v \in L^p([T_0,T_1], W^{s,p}(B_2)) \cap L^{p-1}([T_0,T_1];L_{sp}^{p-1}({\mathbb {R}}^n)) \cap C([T_0,T_1];L^2(B))\) be a weak solution to
In addition, assume that F is a globally Lipschitz function with \(F(0)=0\), which is either bounded or \(F(a)=a\). Then we have:
where \({\mathcal {F}}(a):= \int _{0}^{a} f(t) \;\textrm{d}t\) is the primitive function of F.
Proof
The proof is essentially the same as [7, Lemma 3], except that here we do not use a cutoff function and do not have the global boundedness of u in the ball. For simplicity, we assume \(x_0= 0\), \(R=1\) and \(\sigma =2\).
For a function \(\varphi \in C((T_0,T_1); L^2(B)) \cap L^p((T_0,T_1); X^{s,p}_0(B,B_2))\), we use the following regularization of functions
where \(\zeta (\sigma )\) is a smooth function with compact support in \((-\frac{1}{2}, \frac{1}{2})\) satisfying
This regularization process gives us a test function \(\varphi ^\varepsilon \in C^1((T_0+ \varepsilon ,T_1- \varepsilon ); L^2(B)) \cap L^p((T_0+\varepsilon ,T_1-\varepsilon ); X^{s,p}_0(B,B_2) )\). Let \(t_0 = T_0 + \varepsilon _0\) and \(t_1 = T_1- \varepsilon _0\) and we test the equation with \(\varphi ^\varepsilon \) as above, for \(\varepsilon < \frac{\varepsilon _0}{2}\). First, we will show the claim for the smaller interval \([t_0,t_1] \subset [T_0,T_1]\), and then through a limiting argument, prove the result for the whole interval. As in equation (3.5) in [7], we get
and we obtain a similar identity for v without \(\int _{t_0}^{t_1} \int _{B} \varphi ^\varepsilon f \;\textrm{d}x \;\textrm{d}t\) in the right-hand side. Here \(\Sigma _u\) is defined by
Observe that by using an integration by parts, the term \(\Sigma _u(\varepsilon )\) can be rewritten as
where we also used that \(\zeta \) has compact support in \((-1/2,1/2)\). By subtracting the identities for u and v, we obtain
Now we take \(\varphi \) to be \(F(u^\varepsilon -v^\varepsilon )\). Observe that
After an integration by parts, we get
We now wish to pass to the limit in \({\mathcal {I}}_1, {\mathcal {I}}_2\) and \({\mathcal {I}}_3\). Let \(w=u-v\), we now treat \({\mathcal {I}}_1\). The fact that F is globally Lipschitz together with \(F(0)=0\) implies \(|F(t)| \le C |t|\). Therefore,
where C is the Lipschitz constant of F. After integrating and using Hölder’s inequality, we obtain
Since \(w^\varepsilon \in C((T_0+ \varepsilon _0, T_1-\varepsilon _0); L^2(B))\), uniformly, we have
Observe that
which tends to zero since w is in \(C([T_0,T_1],L^2(B))\). In a similar way, one can argue that
Using the triangle inequality, we get
using a computation similar to (6.3), we obtain
and
These two expressions converge to zero, since \(w \in C([T_0,T_1],L^2(B))\) and \((t_0-\varepsilon ,t_1+\varepsilon )\Subset (T_0,T_1)\). This shows that \({\mathcal {I}}_1\) converges to zero. In a similar way, one can argue that \({\mathcal {I}}_2\) tends to zero. For the term \({\mathcal {I}}_3\), we have
The sequence \(w^\varepsilon \) is bounded in \(L^{p_s^\star ,p}(B\times (t_0,t_1))\); therefore, it has a weakly convergent subsequence. Using the pointwise convergence of \(w^\varepsilon \) to w, we get the weak convergence of \(w^\varepsilon - w\) to zero. By the assumptions on q, r together with Hölder’s inequality (2.11), f(x, t) belongs to the dual space \(L^{(p_s^\star )^\prime ,p^\prime }(B\times (t_0,t_1))\). Therefore,
On the other hand,
Recall that the shift operator,
for a function \(g \in L^p(t_0-\varepsilon _0,t_1+\varepsilon _0)\) is continuous for \(-\varepsilon _0 \le a \le \varepsilon _0\). Hence, we get
Upon passing to a subsequence \(w^\varepsilon (x,t+\varepsilon \sigma )\) and \(w^\varepsilon (x,t)\) converge weakly in \(L^{p_s^\star ,p}(B \times (t_0,t_1))\), since they converge to w(x, t) pointwise, we get the weak convergence
Combined with the convergence of the norms, this implies the strong convergence in the norm; in particular, we have
Now we turn our attention to the terms on the left-hand side of (6.2). The terms \(\Sigma _u(\varepsilon )\) and \(\Sigma _v(\varepsilon )\) converge to zero. To show this, we start with the following computation, borrowed from [7, Lemma 3.3]. Using a suitable change of variables in (6.1) and recalling \(\varphi =F(w^\varepsilon )\), we can also write
In a similar way to the argument for convergence of \({\mathcal {I}}_1\), we can see that
We spell out the details of the arguments for convergence of \(\Sigma _u^2(\varepsilon )\).
where C is the Lipschitz constant of F. We have used \(|\zeta | \le 1\) and \(|\zeta ^\prime | \le 8\) in the computation. Since \(u\in C([T_0,T_1];L^2(B))\), we get
Using a computation similar to (6.3), we obtain
This converges to zero since \(w \in C([T_0,T_1];L^2(B))\), and \((t_0-\varepsilon ,t_1+\varepsilon ) \Subset (T_0,T_1)\) due to the choice of \(\varepsilon \). In conclusion,
In a similar way, one can argue that
Hence, \(\lim _{\varepsilon \rightarrow 0} \Sigma _u(\varepsilon )=0\). The treatment of \(\Sigma _v(\varepsilon )\) is similar.
The term
converges to
To show this, we consider two cases.
Case A: F is bounded. In this case, \({\mathcal {F}}\) is globally Lipschitz, that is, \( |{\mathcal {F}}(a) - {\mathcal {F}}(b)| \le C |a-b|\); therefore,
which converges to zero as was explained before, see (6.4).
Case B: In this case, we have \({\mathcal {F}}(a)=a^2\). Therefore,
and since \(w \in C([T_0,T_1];L^2(B))\), with an argument similar to the treatment of \({\mathcal {I}}_1\), as we let \(\varepsilon \) go to zero this term converges to zero.
Now we discuss the convergence of the nonlocal term. Our treatment is similar to the argument in [7, Appendix B]. The aim is to show that the following converges to zero.
We split it into the two parts
Here we have used the boundary condition \(u=v (w=0) \) for \(y \in {\mathbb {R}}^n {\setminus } B\). Since \(|F(a)-F(b)| \le C |a-b|\), we have
After passing to a subsequence this sequence converges weakly in \(L^p((t_0,t_1);W^{s,p}(B_2))\) to F(w(x, t)) or in another words
converges weakly in \(L^p \bigl ((t_0,t_1);L^p(B_2 \times B_2) \bigr )\), and since
belongs to \(L^{p^\prime } \bigl ((t_0,t_1); L^{p^\prime }(B_2 \times B_2) \bigr )\), we get the desired convergence for \(\Theta _1(\varepsilon )\). Now for \(\Theta _2(\varepsilon )\) consider
Then for almost every \(x \in B \),
The terms \(|u(x,t)|^{p-1}\) and \(|v(x,t)|^{p-1}\) belongs to \(L^{p^\prime }\bigl ((t_0,t_1); L^{p^\prime }(B)\bigr )\) since \(u,v \in L^p((t_0,t_1); L^p(B))\). The tail term its independent of x and belongs to \(L^{p^\prime }(t_0,t_1)\) by the assumption
Thus, \({\mathcal {G}}(x,t) \in L^{p^\prime }([T_0,T_1]; L^{p^\prime }(B_2))\) and as before after extracting a subsequence:
This shows that
converges to zero.
Finally, we let \(\varepsilon _0\) go to zero to get the desired result for \([T_0,T_1]\). We need to show that the following converge to zero as \(\varepsilon _0\) tends to 0.
and
and
The arguments will be reminiscent of the ideas in the previous part.
We start with \({\mathcal {J}}_2\), in the case of a bounded F, \({\mathcal {F}}\) is globally Lipschitz and we have
This converges to 0 since \(w \in C([T_0,T_1];L^2(B))\), in the case of \(F(a)=a\), we have
Again since \(w \in C([T_0,T_1];L^2(B))\), this term converges to 0.
\({\mathcal {J}}_1\) can be treated in a similar way. For the term \({\mathcal {J}}_4\), using \(|F(a)| \le C |a|\) we get
Since \(w \in L^{p_s^\star ,p}(B \times [T_0,T_1])\) and \(f \in L^{(p_s^\star )^\prime ,p^\prime }(B \times [T_0,T_1])\), using Hölder’s inequality (2.11), one can see that
Now using the absolute continuity of the integral for integrable functions, we can conclude that \({\mathcal {J}}_4\) converges to 0. The reasoning for convergence of \({\mathcal {J}}_3\) is similar.
Now we turn our attention to the nonlocal terms.
First, we treat \(\Theta _1\). Notice that since \(u,v \in L^p([T_0,T_1];W^{s,p}(B_2))\) we have
and using Lipschitz continuity of F and the fact that \(w \in L^p([T_0,T_1];W^{s,p}(B_2))\), we have
This implies that the integrand involved in \(\Theta _1\) belongs to \(L^1([T_0,T_1];L^1(B_2\times B_2))\). And similar to the treatment of \({\mathcal {J}}_4\), since the volume of the integration region is shrinking to 0, \(\Theta _1\) converges to 0. To deal with \(\Theta _2\), notice that
and define
We can estimate this integration in terms of the tail, that is,
see, for example, (6.6). Therefore, \({\mathcal {G}}(x,t) \in L^{p^\prime }([T_0,T_1]; L^{p^\prime }(B))\). Hence, using Hölder’s inequality
This concludes the result. \({\mathcal {N}}_1\) can be treated in an exactly similar manner. \(\square \)
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Tavakoli, A. A perturbative approach to Hölder continuity of solutions to a nonlocal p-parabolic equation. J. Evol. Equ. 24, 27 (2024). https://doi.org/10.1007/s00028-024-00949-8
Accepted:
Published:
DOI: https://doi.org/10.1007/s00028-024-00949-8