Existence for doubly nonlinear fractional p-Laplacian equations

We prove the existence of a global-in-time weak solution to a doubly nonlinear parabolic fractional p-Laplacian equation, which has general double nonlinearity including not only the Sobolev subcritical/critical/supercritical cases but also the slow/homogenous/fast diffusion ones. Our proof exploits the weak convergence method for the doubly nonlinear fractional p-Laplace operator.

1. Let \({\mathcal {X}}\) be a Banach space. A function \([0,\infty ) \ni t \mapsto u(t)\equiv u(\cdot , t) \in {\mathcal {X}}\) is called weakly continuous in \({\mathcal {X}}\) provided that \( [0,\infty ) \ni t\mapsto \langle u(t), \psi \rangle _{{\mathcal {X}} \times {\mathcal {X}}^\prime }\) is continuous for every \(\psi \in {\mathcal {X}}^\prime \). Here \(\langle \cdot , \cdot \rangle _{{\mathcal {X}} \times {\mathcal {X}}^\prime }\) represents the dual pairing between \({\mathcal {X}}\) and \({\mathcal {X}}^\prime \).

The authors are grateful to the referees for the careful reading of the original version of the manuscript and for the many sharp comments that eventually led to an improvement of presentation; in particular, the referee pointed out Remark 5.2. Moreover, MM acknowledges the partial support by the Grant-in-Aid for Scientific Research (C) Grant No.21K03330 (2021) at JSPS. The work by KN was also partially supported by Grant-in-Aid for Young Scientists Grant No.21K13824 (2021) at JSPS. YY was also partially supported by Grant-in-Aid for Scientific Research (C) Grant No.18K03381 (2018) at JSPS, and by Individual Research Expense of College of Humanities and Sciences at Nihon University for 2020.

Authors and Affiliations

1. Department of Mathematics, Nippon Institute of Technology, Saitama, Japan

   Nobuyuki Kato

2. Faculty of Advanced Science and Technology, Kumamoto University, Kumamoto, 860-8555, Japan

   Masashi Misawa

3. Headquarters for Admissions and Education, Kumamoto University, Kumamoto, 860-8555, Japan

   Kenta Nakamura

4. Department of Mathematics, College of Humanities and Sciences, Nihon University, Tokyo, Japan

   Yoshihiko Yamaura

Corresponding author

Correspondence to Kenta Nakamura.

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Appendix A: Proof of Lemma 3.1

In this appendix, we prove the existence of solutions of (3.1), Lemma 3.1. The proof is based on the direct method in the calculus of variations. In the course of this appendix, let \(s \in (0,1)\), \(p>1\) and \(q>0\) be given and set \({\mathbb {X}}:=W_0^{s,p}(\Omega ) \cap L^{q+1}(\Omega )\).

Proof of Lemma 3.1

Given \(u_{m-1} \in {\mathbb {X}}\), we define the functional on \({\mathbb {X}}\):

The proof is twofold: first, we shall show the existence of a minimizer of the functional \({\mathcal {F}}(w)\), then we shall demonstrate the first variation of \({\mathcal {F}}(w)\), both of them show the validity of Lemma 3.1.

By Young’s inequality, there holds for any \(w \in {\mathbb {X}}\)

Let \(\{u_k\}_{k=1}^\infty \) be a minimizing sequence, such that

for all \(k \in {\mathbb {N}}\), where the infimum is taken for \(w \in {\mathbb {X}}\). Note that, for all \(w \in {\mathbb {X}}\),

with some \(v \in {\mathbb {X}}\) not empty, because \(C^\infty _0 (\Omega ) \subset {\mathbb {X}}\). We then observe that

and, by (A.1),

implying

Up to not relabelled subsequences, there exists a weak limit \(u \in L^{q+1}(\Omega )\) such that

and thus, by the Banach-Steinhaus theorem and (A.3),

Since \(u_k \in W^{s,p}_0(\Omega )\), by the fractional Poincaré inequality (2.13) in Lemma 2.3 and (A.3)\(_2\) we have, \(\sup \limits _{k \in {\mathbb {N}}}\Vert u_k\Vert _{L^p({\mathbb {R}}^n)}<\infty \), in turn yielding with (A.3)

Letting \(s^\prime \equiv \min \{\frac{n}{2p},s\}\) implies \(0<s^\prime \leqslant s <1\). By Lemma 2.4,

and, for any ball \(B_R\) satisfying \(\Omega \subset B_R\)

Since \(s^\prime p \leqslant \frac{n}{2} <n\), we can apply Lemma 2.6 to get (after passing to a subsequence, if necessary)

for all r with \(1\leqslant r<\frac{np}{n-s^\prime p}\) and

which, in particular, yields \(u=0\) in \(\Omega ^c\). From Fatou’s Lemma with (A.7) and (A.6), it follows that

Consequently, by (A.5) and (A.8), it ensures that \(u \in {\mathbb {X}}\). Moreover, thanks to the weak convergence (A.4) we have

this and the two displays (A.5) and (A.8) imply that

which assures that \(u \in {\mathbb {X}}\) is actually a minimizer of \({\mathcal {F}}\). Finally, a completely standard argument shows that \(\{u_m\}_{m \in {\mathbb {N}}}\) satisfies the Euler–Lagrange equation

finishing the proof of Lemma 3.1.

Appendix B: Boundedness

In this Appendix B, through the additional stronger assumption of boundedness of the initial datum \(u_0\), we conclude slightly stronger assertions compared to ones in the previous section (see Lemmas B.1 and B.2 and Theorem B.3).

1.1 B.1 Boundedness

In this Appendix B.1, we show the boundedness for approximating solutions of (1.1). The test function here is retrieved from [3].

Lemma B.1

(Boundedness) Suppose that \(u_0 \in W^{s, p}_0 (\Omega ) \cap L^\infty (\Omega )\). Let \({\bar{u}}_h\), \(u_h\),  \({\bar{w}}_h\),  \(w_h\),  \(v_h\) and \({\bar{v}}_h\) be approximate solutions of (1.1) defined in (3.3) and (3.4). Then, these solutions are bounded in \(\Omega _\infty \):

Proof

The estimates (B.1)\(_2\), (B.2) and (B.3) simply follow from (B.1)\(_1\) and Lemma 3.2. Following the argument developed in [27, Lemma 3.3, p. 164], we shall prove  (B.1)\(_1\). Put \(M:=\Vert u_0\Vert _{L^\infty (\Omega )}\) for brevity. In (3.2), choose a test function \(\xi _\delta (u_m)\) as

Note that \(\xi _\delta (\sigma )\) is clearly bounded and Lipschitz function for \(\sigma \in {\mathbb {R}}\) and thus, \(\xi _\delta (u_m) \in W^{s, p}_0(\Omega ) \cap L^\infty (\Omega )\).

Then, for \(m=1,2,\ldots \), we have

The integrand in the fractional integral of (B.4) is evaluated as follows: Since \(\xi _\delta (u_m(x))-\xi _\delta (u_m(y))\) has the same sign as \(u_m(x)-u_m(y)\) via the monotonicity of \(\xi _\delta (\sigma )\), we observe

on \({\mathbb {R}}^n \times {\mathbb {R}}^n\).

Dropping the fractional term of (B.4) implies

Thanks to the dominated convergence theorem and passing to the limit \(\delta \searrow 0\) in (B.5) we have

which is equivalent to

By neglecting the region where \(|u_{m-1}|\leqslant M \leqslant |u_m|\), the right-hand side is further estimated as

Summarizing, we have

Iterating the above display with respect to m, we obtain

for all \(m=1,2,\ldots \), which in turn implies \(|u_{m}|\leqslant M\) in \(\Omega \) for any \(m=1,2,\ldots \), finishing the proof. \(\square \)

Graph of \(y=\xi _\delta (\sigma )\)

Full size image

By the boundedness (Lemma B.1), the convergence result in Lemma 4.7 improves as follows:

Lemma B.2

(Convergence of approximate solutions) Suppose that \(u_0 \in W^{s, p}_0 (\Omega ) \cap L^\infty (\Omega ).\) Let \({\bar{u}}_h\),  \(u_h\),  \({\bar{w}}_h\),  \(w_h\), \({\bar{v}}_h\) and \(v_h\) be the approximate solutions of (1.1) defined by (3.3) and (3.4), satisfying the convergence in Lemma 4.7. For all finite exponent \(\gamma \geqslant 1\), any bounded domain \(K \subset {\mathbb {R}}^n\) and every positive number \(T<\infty \),

as \(h\searrow 0\).

Proof

From the boundedness (B.1) and (4.20) it follows that

The boundedness (B.1), (B.10) and the convergence (4.21) with the Hölder inequality gives (B.7)\(_1\). The validity of other statements (B.7)\(_2\), (B.8) and (B.9) is shown by use of the boundedness Lemma B.1, (4.29) and (4.30). \(\square \)

1.2 B.2 A regularity of \(\varvec{\partial _t\left( |u|^{q-1}u\right) }\) in the case \(q \geqslant 1\)

Here we state additional regularity for the time-derivative, obtained from the boundedness Lemma B.1.

Theorem B.3

Let \(q \geqslant 1\). Suppose that the initial datum \(u_0 \in W^{s, p}_0 (\Omega ) \cap L^\infty (\Omega )\). Then, the weak solution u of (1.1) obtained in Theorem 1.1 satisfies \(\partial _t (|u|^{q - 1} u) \in L^2 (\Omega _\infty )\).

In order to prove Theorem B.3, we will derive the \(L^2\)-estimate of the time-derivative:

Lemma B.4

Let \(q \geqslant 1\) and \(v_h\) be the approximate solutions of (1.1) defined by (3.4). Assume that \(u_0 \in W^{s, p}_0 (\Omega ) \cap L^\infty (\Omega )\). Then the time derivatives of approximate solutions \(\{\partial _t v_h\}\) are bounded in \(L^2 (\Omega _T)\) for any positive \(T < \infty \),

with a positive constant \(C=C(p,q)\).

Proof of Lemma B.4

From the energy estimate (3.12) in Lemma 3.3 and (B.1) in Lemma B.1, we obtain

which finishes the proof of (B.11).

We report the proof of Theorem B.3.

Proof of Theorem B.3

Let \(q \geqslant 1\). By (B.11) in Lemma B.4, \(\{\partial _t v_h\}_{h>0}\) is bounded in \(L^2 (\Omega _T)\) and thus, there are a subsequence \(\{\partial _t v_h\}_{h>0}\) (also labelled with h) and a limit function \(\vartheta \in L^2(\Omega _T)\) such that, as \(h \searrow 0\),

By the convergence (B.9) in Lemma B.2 we pass to the limit as \(h \searrow 0\) in the identity

for every \(\varphi \in C^\infty _0(\Omega _T)\) to obtain from (4.37) that

As a consequence, by the convergence (B.9) in Lemma B.2 with (B.11) in Lemma B.4, there holds that

Since the right-hand side is independent of T, we are allowed to pass to the limit \(T \nearrow \infty \) in the last display, which finishes the proof of Theorem B.3. \(\square \)

Appendix C: A space-time fractional Sobolev inequality

In this appendix, we accomplish in Lemma C.1 a general inequality interpolating by the Gagliardo seminorm in time-space domain in the special case where the function is weakly differentiable with respect to time. The inequality in Lemma C.1 is used essentially as the key ingredient for Lemma 4.3

The following inequality gives an interpolation between fractional Sobolev semi-norms and the integral of time derivative.

Lemma C.1

(Space-time fractional Sobolev inequality) Let \(K \subset {\mathbb {R}}^n\) be a bounded domain and let \(I:=(0,T)\) with \(0<T<\infty \). Let \(s^\prime , {\bar{s}}\) be two exponents obeying \(0<s^\prime<{\bar{s}} <1\). Then it holds

where the integrals appearing in the right-hand side are assumed to be finite.

Proof

To begin, we divide the Gagliardo semi-norm into two terms:

By the assumption it holds \(1+(s^\prime -{\bar{s}}) >0\), thereby giving

Due to the fundamental theorem of calculus,

this together with (C.2) and Fubini’s theorem gives

Here, via the change of variable \(\varrho =|x-x'|\) we have

Similarly, by the condition \(0<{\bar{s}}<1\), the integral \(\int _{-T}^T\frac{1}{|\tau |^{{\bar{s}}}} d\tau \) appearing in (C.4) takes a finite value depending T only. Combining this with (C.4) and (C.5), we get

On the other hand, the integration \((\textrm{II})\) is estimated as

where, in the penultimate line, we computed

Finally, merging (C.6),  (C.7) in (C.3), we end up with the conclusion (C.1). \(\square \)

Appendix D: Proof of Lemma 5.1

This appendix is devoted to the proof of Lemma 5.1.

Proof of Lemma 5.1

At the beginning, we verify that \([0,\infty ) \ni t \mapsto v_h (t)\) is (uniformly in h) weakly continuous in \(L^{\frac{q+1}{q}} (\Omega )\)^{Footnote 1}. Let \(t_2>t_1 \geqslant 0\) be arbitrarily taken. Testing the approximating equation (3.5) with \(\psi (x)\text{1 }\hspace{-0.25em}\text{ l}_{(t_1,t_2)}(t)\), where \(\psi =\psi (x) \in C^\infty ({\mathbb {R}}^n)\) with \(\textrm{supp}\,(\psi ) \subset \Omega \) and \(\text{1 }\hspace{-0.25em}\text{ l}_{(t_1,t_2)}(t)\) is the usual Lipschitz approximation of characteristic function of the time-interval \((t_1, t_2)\). By using of Hölder’s inequality and  (3.13) in Lemma 3.3, it holds that

whenever \(t_2>t_1 \geqslant 0\). By the density of \(C^\infty _0 (\Omega )\) in \(L^{q+1} (\Omega )\), Hölder’s inequality and (3.13) in Lemma 3.3 again, the above display implies that \([0,\infty ) \ni t \mapsto v_h (t)\) is (uniformly in h) weakly continuous in \(L^{\frac{q+1}{q}} (\Omega )\), as desired.

Next, for \(\varepsilon >0\) small enough, we define

In the weak formulation (3.5) testing with \(\zeta _\varepsilon (t)\varphi \), where \(\varphi \in C^\infty ({\mathbb {R}}^n \times [0,T))\) has support contained in \(\Omega \times [0,T)\), we get

By definition it holds that \(v_h(0)=|u_0(x)|^{q-1}u_0(x)\) and, as shown above, \([0,\infty ) \ni t \mapsto v_h(t)\) is weakly continuous in \(L^{\frac{q+1}{q}}(\Omega )\), thereby taking the limit \(\varepsilon \searrow 0\) in the last display yields

therefore, thanks to convergences (4.37) in Lemma 4.8 and (5.3) in Step 2 of the proof of Theorem 1.1, sending \(h \searrow 0\) renders that

as wanted. \(\square \)

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