Abstract
In this paper, we study the degenerate parabolic variational inequalities in a bounded domain. By solving a series of penalty problems, the existence and uniqueness of the solutions in the weak sense are proved by the energy method and a limit process.
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1 Introduction
Let \(0 < T < \infty \) and \(\Omega \subset {{\mathrm{{R}}}_{N}} (N \ge 2)\) be a bounded simple domain with appropriately smooth boundary ∂Ω. In this article, we consider the following quasilinear degenerate parabolic inequalities:
with
where \({Q_{T}} = \Omega \times (0,T]\), \({\mathrm{{a}}}(u) = {u^{\sigma }} + {d_{0}}\), and \({\Gamma _{T}}\) is the lateral boundary of cylinder \(Q_{T}\).
In applications, Problem (1.1) arises in the model of American option pricing in the Black–Scholes models. We refer to [1–4] for the financial background of parabolic inequalities. Among them, the most interesting research topic is to construct different types of variational parabolic inequalities and analyze the existence and uniqueness for their solutions (see, for example, [3–10] and the references therein). In 2014, the authors in [5] discussed the problem
with second-order elliptic operator
They proved the existence and uniqueness of a solution to this problem with some restrictions on \(u_{0}\), F, and L. Later, the authors in [6, 7] extended the relative conclusions with the assumption that \(a(u)\) is a constant, and \(p(x) =2\). The authors discussed the existence and numerical algorithm of the solution.
To the best of our knowledge, the existence and uniqueness of this problem with the assumption that \(p(x,t)\) are variables have been less studied. We cannot easily apply the method in [6, 7] to the case that \(p(x,t)\) and \(a(u)\) are not constants.
The aim of this paper is to study the existence and uniqueness of solutions for a degenerate parabolic variational inequality problem. Throughout the paper, we assume that the exponent \(p(x, t)\) is continuous in \(Q_{T}\) with a logarithmic module of a country:
where \({p^{-} } = \mathop {\inf } _{(x,t) \in {Q_{T}}} p(x,t)\) and \({p^{+} } = \mathop {\sup } _{(x,t) \in {Q_{T}}} p(x,t)\).
The outline of this paper is as follows: In Sect. 2, we introduce the function spaces of Orlicz–Sobolev type, give the definition of the weak solution to the problem, and state our main theorems. In Sect. 3, we give some estimates of the penalty problem (approximating problem). Section 4 proves the existence and uniqueness of the solution obtained in Sect. 2.
2 The main results of weak solutions
In this section, we recall some useful definitions and known results, which can be found in [11–14]. Set
and denote by \(W'({Q_{T}})\) the dual of \(W({Q_{T}})\) with respect to the inner product in \({L^{2}}({Q_{T}})\).
In the spirit of [3] and [4], we introduce the following maximal monotone graph
where \(\theta \in [0,M)\) and M depends only on \(|u_{0}|_{\infty}\).
The purpose of the paper is to obtain the existence and uniqueness of weak solutions of (1.1). Let \(B = W({Q_{T}}) \cap {L^{\infty }}(0,T;{L^{\infty }}(\Omega ))\), and the weak solution is defined as:
Definition 2.1
A pair is called a weak solution of problem (1.1), if (a) \(u(x,t) \ge {u_{0}}(x)\), (b) \(u(x,0) = {u_{0}}(x)\), (c) \(\xi \in G(u - {u_{0}})\), (d) for every test function \(\phi \in Z \equiv \{ \eta (z):\eta \in W({Q_{T}}) \cap {L^{\infty }}(0,T;{L^{2}}( \Omega )),{\eta _{t}} \in W({Q_{T}})\} \) and every \({t_{1}},{t_{2}} \in [0,T]\) the following identity holds:
Our main results are the following two theorems.
Theorem 2.1
Let us satisfy conditions (1.3). If the following conditions hold:
-
(H1)
\(\max \{ 1,\frac{{2N}}{{N + 1}}\} < {p^{-} } < N\), \(2 \le \sigma < \frac{{2{p^{+} }}}{{{p^{+} } - 1}}\), \(0 < \gamma < {d_{0}}\), and
-
(H2)
\({u_{0}}(x) \ge 0, f \ge 0, { \Vert {{u_{0}}} \Vert _{ \infty,\Omega }} + \int _{0}^{T} {{{ \Vert {f(x,t)} \Vert }_{ \infty,\Omega }}{\,\mathrm{d}}t} + \vert \Omega \vert \cdot T = K(T) < \infty \),
then Problem (1.1) has at least one weak solution in the sense of Definition 2.1.
Theorem 2.2
Suppose that the conditions in Theorem 2.1are fulfilled and \(p^{+} \geq 2\). Then, Problem (1.1) admits a unique solution in the sense of Definition 2.1.
3 Penalty problems
In this section, we consider a family of auxiliary parabolic problems
with
\({\beta _{\varepsilon }}( \cdot )\) is the penalty function satisfying
With a similar method as in [8], we may prove that the regularized problem has a unique weak solution
satisfying the following integral identities
and
We start with two preliminary results that will be used several times below.
Lemma 3.1
Let \(M(s) = { \vert s \vert ^{p(x,t) - 2}}s\), then \(\forall \xi,\eta \in {{\mathrm{{R}}}^{N}}\)
Proof
The proof can be found in [15]. □
Lemma 3.2
(Comparison principle)
Assume \(2 < \sigma < \frac{{2{p^{+} }}}{{{p^{+} } - 1}}\), \({p^{+} } \ge 2\), u and v are in \(W({Q_{T}}) \cap {L^{\infty }}(0,T;{L^{ \infty }}(\Omega ))\). If \({L_{\varepsilon }}u \ge {L_{\varepsilon }}v\) in \(Q_{T}\) and \(u(x,t) \le v(x,t)\) on \(\partial {Q_{T}}\), then \(u(x,t) \le v(x,t)\) in \(Q_{T}\).
Proof
We argue by contradiction. Suppose \(u(x,t)\) and \(v(x,t)\) satisfy \({L_{\varepsilon }}u \ge {L_{\varepsilon }}v \) in \({Q_{T}}\) and there is a \(\delta > 0\) such that for \(0 < \tau \le T\), \(w = u - v\) on the set
and \(\mu ({\Omega _{\delta }}) > 0\). Let
where \(\delta > {\mathrm{{2}}}\varepsilon > 0\) and \(\alpha = \frac{\sigma }{2}\). Let a test function \(\xi = {F_{\varepsilon }}(w) \in Z\) in (3.4),
where \({Q_{T,\varepsilon }} = \{ (x,t) \in {Q_{T}}|w > \varepsilon \} \),
Now, let \({t_{0}} = \inf \{ t \in (0,\tau ]:w > \varepsilon \}\), then we estimate \(J_{1}\) as follows
Let us first consider the case \({p^{-} } \ge 2\). By virtue of the first inequality of Lemma 3.1, we obtain
Noting that \(\frac{{p(x,t)}}{{p(x,t) - 1}} \ge \frac{{p + }}{{p + - 1}} \ge { \sigma ^{2}} = \alpha > 1\) and applying Young’s inequality, we may estimate the integrand of \(J_{3}\) in the following way
Substituting (3.10) into \(J_{3}\) and combining it with \(J_{2}\), we obtain
Recall that \(0 < \gamma \le {d_{0}}\), \(u \in W({Q_{T}}) \cap {L^{\infty }}(0,T; {L^{\infty }}( \Omega ))\). Then, we have
where C is a positive constant. Thus, we insert the above estimates (3.8), (3.9), (3.11), and (3.12) into (3.7) and dropping the nonnegative terms, we arrive at
Secondly, we consider the case \(1 < {p^{-} } \le p(x,t) < 2, {p^{+} } \ge 2\). According to the second inequality of Lemma 3.1, it is easily seen that the following inequalities hold
Substituting the above inequality into \(J_{3}\), we obtain
Similar to the case \({p^{-} } \ge 2\), estimate (3.13) still holds using (3.14) instead of (3.11). Note that \(\mathop {{\mathrm{{lim}}}} _{ \varepsilon \to 0} (\delta - {\mathrm{{ 2}}}\varepsilon )({\mathrm{{1 }}} - {{\mathrm{{2}}}^{{\mathrm{{1}}} - \alpha }}){\varepsilon ^{{\mathrm{{1}}} - \alpha }}\mu ({\Omega _{\delta }}) = + \infty \), we obtain a contradiction. This means \(\mu ({\Omega _{\delta }}) = 0\) and \(w \le 0\) a.e. in \({{\mathrm{{Q}}}_{\tau }}\). □
Lemma 3.3
Let \({u_{\varepsilon }}\) be weak solutions of (3.1). Then,
where \(|{u_{0}}{|_{\infty }} = \mathop {\sup } _{x \in \Omega } |{u_{0}}(x)|\).
Proof
First, we prove \({u_{\varepsilon }} \ge {u_{0\varepsilon }}\) by contradiction. Assume \({u_{\varepsilon }} \le {u_{0\varepsilon }}\) in \(Q_{T}^{0}\), \(Q_{T}^{0} \subset Q_{T}\). Noting \({u_{\varepsilon }} \ge {u_{0\varepsilon }}\) on \(\partial {Q_{T}}\), we may assume that \({u_{\varepsilon }} = {u_{0\varepsilon }}\) on \(\partial {Q_{T}}\). With (3.1) and letting \(t = 0\), it is easy to see that
From Lemma 3.2, we arrive at
Therefore, we obtain a contradiction.
Secondly, we pay attention to \({u_{\varepsilon }}(t,x) \le { \vert {{u_{0}}} \vert _{\infty }} + \varepsilon \). Applying the definition of \({\beta _{\varepsilon }}( \cdot )\), we have that
From (3.20), we obtain
and \({u_{\varepsilon }}(t,x) \le { \vert {{u_{0}}} \vert _{\infty }} + \varepsilon \) in Ω. Thus, combining (3.20) and (3.21) and repeating Lemma 3.2, we have
Thirdly, we aim to prove (3.16). From (3.1),
It follows by \({\varepsilon _{1}} \le {\varepsilon _{2}}\) and the definition of \({\beta _{\varepsilon }}( \cdot )\) that
Thus, combining the initial and boundary conditions in (3.1) can be proved by Lemma 3.2. □
To prove this theorem, we need the following lemmas.
Lemma 3.4
The solution of problem (3.1) satisfies the estimate
Proof
Let us introduce the following function
The function \(u_{\varepsilon,M}^{2k - 1}\), with \(k \in N\), can be chosen as a test function in (3.4). Let \({t_{2} } = t + h,{t_{1}} = t\) in (3.4), with \(t,t + h \in ( 0, T ) \). Then,
Dividing the last equality by h, letting \(h \to 0\), and applying Lebesgue’s dominated convergence theorem, we have that
By Holder’s inequality, we have
Using Minkowski’s inequality, we arrive at
From (3.15) and the definition of \({\beta _{\varepsilon }}( \cdot )\), we have that
Recall that \(0 < \gamma < {d_{0}}\). Then, we use Lemma 3.1 to find
Substituting (3.29) and (3.30) into (3.28), we arrive at the inequality
Integrating over \((0, t)\) in (3.32) and dropping the nonnegative term (3.31), we arrive at
Then, as \(k \to \infty \), we have that
If we choose \(M > K(T)\), then
and therefore \({u_{\varepsilon,M}}( \cdot,t) = {u_{\varepsilon }}( \cdot,t)\). □
Lemma 3.5
The solution of problem (3.1) satisfies the estimates
Proof
To prove Lemma 3.5, we proceed as in the proof of Lemma 3.4, and in (3.27) we take \(k=1\). We then obtain
Therefore, integrating in time over \((0,t)\), \(\forall t \in (0,T)\),
and since the first and third terms on the left-hand side are nonnegative and recalling the L2-norm
From this we obtain (3.34). Since \(a({u_{\varepsilon }}) \ge {d_{0}}\), \({a_{\varepsilon,M}}({u_{ \varepsilon }}) \ge u_{\varepsilon }^{\sigma }\), (3.35), and (3.35) are immediate consequences of (3.34). □
Lemma 3.6
The solution of problem (3.1) satisfies the estimate
Proof
From identity (3.5), we obtain
where
First, we pay attention to \(A_{1}\). Using Holder inequalities we obtain
When \(\int _{0}^{t} {\int _{\Omega }{{{ ( {a({u_{\varepsilon }}){{ \vert {\nabla {u_{\varepsilon }}} \vert }^{p(x,t) - 1}}} )}^{ \frac{{p(x,t)}}{{p(x,t) - 1}}}}{\,\mathrm{d}}x{\,\mathrm{d}}t} } \ge 1\), we arrive at
Moreover, when \(\int _{0}^{t} {\int _{\Omega }{{{ ( {a({u_{\varepsilon }}){{ \vert {\nabla {u_{\varepsilon }}} \vert }^{p(x,t) - 1}}} )}^{ \frac{{p(x,t)}}{{p(x,t) - 1}}}}{\,\mathrm{d}}x{\,\mathrm{d}}t} } < 1\), we obtain
Combining (3.39) and (3.40), and using Lemma 3.5, we arrive at
Secondly, we calculate \(A_{2}\) and \(A_{3}\). Following a similar procedure as (3.41), we obtain
Substituting (3.41), (3.42), and (3.43) into (3.38), we conclude that
Then, we obtain Lemma 3.6. □
4 Proof of the main results
In this section, we are ready to prove Theorem 2.1 and Theorem 2.2. From (3.15), Lemma 3.5, and Lemma 3.6, we see that \({u_{\varepsilon}}\) is bounded and increasing in ε, which implies the existence of a function u, such that, as \(\varepsilon \to 0\)
for some functions \(u \in W({Q_{T}})\), \({A_{i}}(x,t) \in {L^{p'(x,t)}}({Q_{T}})\), \({W_{i}}(x,t) \in {L^{p'(x,t)}}({Q_{T}})\).
Lemma 4.1
For almost all \((x,t) \in {Q_{T}}\),
Proof
In (4.4) and (4.5), letting \(\varepsilon \to 0\), we have that
By Lebesgue’s dominated convergence theorem, we have
Hence, we have
This completes the proof of Lemma 4.1. □
Lemma 4.2
For almost all \((x,t) \in {Q_{T}}\),
Proof
In (3.5), choosing \(\xi = \Phi \cdot ({u_{\varepsilon }} - u)\) with \(\Phi \in W({Q_{T}})\), \(\Phi \ge 0\), we have
It follows that
On the other hand, from \({u_{\varepsilon }},u \in {L^{\infty }}(Q_{T})\), \(|\nabla u| \in {L^{p(x,t)}}({Q_{T}})\), we obtain
Note that
Then, the proof of Lemma 4.2 is complete. □
Lemma 4.3
As \(\varepsilon \to 0\), we have
Proof
Using (3.15) and the definition of \({\beta _{\varepsilon }}\), we have
Now, we prove \(\xi \in G(u - {u_{0}})\). According to the definition of \(G( \cdot )\), we only need to prove that if \(u({x_{0}}, {t_{0}}) > {u_{0}}({x_{0}})\),
In fact, if \(u({x_{0}},{t_{0}}) > {u_{0}}({x_{0}})\), there exist a constant \(\lambda > 0\) and a δ-neighborhood \({B_{\delta }}({x_{0}},{t_{0}})\) such that if ε is small enough, we have
Thus, if ε is small enough, we have
Furthermore, it follows by \(\varepsilon \rightarrow 0 \) that
Hence, (4.13) holds, and the proof of Lemma 4.3 is complete. □
The proof of Theorem 2.1.
Applying (3.15), (3.16), and Lemma 4.3, it is clear that
thus (a), (b), and (c) hold. The remaining arguments of the existence part are the same as those of Theorem 2.1 in [8] by a standard limiting process. Thus, we omit the details. □
The proof of Theorem 2.2
We argue by contradiction. Suppose \((u,{\xi _{1}})\) and \((v,{\xi _{2}})\) are two nonnegative weak solutions of problem (1.1). Define \(w = u - v\),
and let \(\xi = {F_{\varepsilon }}(w) \in Z\) be a test function in (3.4),
Now, we prove
On the one hand, if \({u_{1}}(x,t) > {u_{2}}(x,t)\), then using (3.16) yields
From (2.1) and (4.18), it is easy to see that
Combining (4.18) and (4.19) and the fact that \(\alpha = \frac{1}{2}\sigma > 1\), (4.16) is obtained.
On the other hand, if \({u_{1}}(x,t) < {u_{2}}(x,t)\), it is easy to see that \(F(w) = 0\). Equation (4.16) still holds.
Using (4.16) in (4.15) and dropping the nonnegative term, (4.15) becomes
By the above inequality and combining the initial and boundary conditions in (1.1), the uniqueness of the solution can be proved following the similar proof of (3.7)–(3.14). □
5 Conclusion
In this paper, an initial Dirichlet problem of degenerate parabolic variational inequalities in the following form
is studied. The existence and uniqueness of the solutions in the weak sense are proved by the energy method and a limit process. The localization property of weak solutions is also discussed.
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Acknowledgements
The authors are sincerely grateful to the reviewer and the Associate Editor who handled the paper for their valuable comments.
Funding
This work was supported by the Guizhou Provincial Education Foundation of Youth Science and Technology Talent Development (No. [2016]168), the Foundation of Guizhou Minzu University (NO.GZMUSK[2021]ZK02) and the Doctoral Project of Guizhou Education University (NO.2021BS037).
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YDS was a major contributor in writing the manuscript. TW performed the validation and formal analysis. All authors read and approved the final manuscript.
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Sun, Y., Wu, T. Study of weak solutions for degenerate parabolic inequalities with nonstandard conditions. J Inequal Appl 2022, 141 (2022). https://doi.org/10.1186/s13660-022-02872-3
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DOI: https://doi.org/10.1186/s13660-022-02872-3