Hypercontractivity, supercontractivity, ultraboundedness and stability in semilinear problems
-
Davide Addona
Abstract
We study the Cauchy problem associated to a family of nonautonomous semilinear equations in the space of bounded and continuous functions over
1 Introduction
This paper is devoted to continuing the analysis started in [5]. We consider
a family of linear second-order differential
operators
where I is either an open right halfline or the whole
where
We refer the reader to [9] for the construction of the evolution operator
Classical arguments can be adapted to our case to prove the existence of a unique local mild solution
Under reasonable assumptions such a mild solution
for any
for any
As in the linear case we are also interested to set problem (1.2) in an
When they exist, evolution families of measures are in general infinitely many, even the tight ones, where, roughly speaking, tight means that all the measures of the family are essentially concentrated on the same large ball
(see Section 2 for a rigorous definition of tightness).
Under additional assumptions on the coefficients of the operator
Formula (1.5) and the density of
In view of these facts, it is significant to extend
We stress that the first condition on ψ may seem too restrictive, but in fact it is not. Indeed, the Sobolev embedding theorems fail to hold, in
general, when the Lebesgue measure is replaced by any of the measures
Under the previous assumptions, for any
Since, as it has been stressed, in this context the Sobolev embedding theorems fail to hold in general, the summability improving properties of the nonlinear evolution operator
For what concerns the ultraboundedness of
Finally, assuming that
To get the exponential stability of the trivial solution in
Notations
For
where
We use the symbols
The open ball in
2 Assumptions and preliminary results
Let
Hypotheses 2.1.
Our standing assumptions on the coefficients of the operators
The coefficients
For every
with positive infimum
for any
There exists a nonnegative function
for any
There exists a locally bounded function
for any
There exists a function
for any
(2.1)
Under Hypotheses 2.1 (i)–(iii) (actually even under weaker assumptions) it is possible to associate an evolution operator
where
for any
for any
for any
Remark 2.2.
The case
Under Hypotheses 2.1 we can also associate an evolution system of measures
There exists a constant
For more details and the proofs of the results that we have mentioned, we refer the reader to [9, 10, 11, 13].
3 The semilinear problem in a bounded time interval
Given
Hypotheses 3.1.
Our standing assumptions on ψ are as follows.
The function
(3.1)for any
The function
Theorem 3.2.
Under Hypotheses 2.1 and 3.1, for any
where
Finally, if
Proof.
Even if the proof is quite standard, for the reader’s convenience we provide some details.
Fix
Step 1. We prove that there exists
there exists a mild solution to problem (1.2) defined in the time interval
for any
It is also straightforward to see that Γ maps
As a consequence, Γ has a unique fixed point in
Step 2. We prove the uniqueness of the mild solution
for
Using (3.5), we estimate the last two integral terms in the right-hand side of (3.6), which we denote by
The same arguments show that
From (3.5) and (3.8) we conclude that
The generalized Gronwall lemma (see [7]) yields
Step 3. We prove (3.2) and (3.3).
Since
Step 4. We prove that
for some constant
Corollary 3.3.
In addition to the assumption of Theorem 3.2 suppose that
there exist
for any
Proof.
Fix
for any
Remark 3.4.
Suppose that (3.1) is replaced by the condition
in
We now provide some sufficient conditions for the mild solution to problem (1.2) to exist in the large. Such conditions will be crucial to define the nonlinear evolution operator associated with the Cauchy problem (1.2).
Hypotheses 3.5.
We introduce the following assumptions.
For any
for any
For any
In the rest of this section, for any
Theorem 3.6.
Assume that Hypotheses 2.1, 3.1
(ii), 3.5 and condition (3.9) are satisfied. Then, for any
for every
Proof.
We split the proof into two steps.
Step 1. We prove that, for any
To prove that
for any
in
Repeating the same arguments with
Step 2. Fix
on smooth functions v, where ω
is a positive constant to be chosen later on and to fix the ideas we assume that
for any
To estimate the integral terms in the last side of (3.13), we fix
Hence, minimizing over
where
and estimate (3.11) follows at once.
Estimate (3.10) can be proved likewise. Hence, the details are omitted. ∎
As a consequence of Theorem 3.6 we prove the existence of a mild solution to problem (1.2) in the time domain
holds true in
Corollary 3.7.
Under all the assumptions of Theorem 3.6, for any
for any
Proof.
Fix
Next, using (3.11) we can estimate
and
for any
Let us now prove the last part of the statement. We again use an approximation argument.
Fix
first for any
with equivalence of the corresponding norms, by an interpolation argument and (3.16) we deduce
that
for any
Similarly,
Using these estimates, we can now show that
and
for any
and, using (3.18),
From these two estimates we conclude that
for any
and, consequently,
for any
for any
for any
in
The arguments in the proof of Theorem 3.6 and Corollary 3.7 allow us to prove the following result.
Proposition 3.8.
Under Hypotheses 2.1, the following properties are satisfied.
Let
Let
Proof.
To prove property (i), it suffices to apply the Banach fixed point theorem in the space of all the functions
To prove property (ii), one can argue by approximation. We fix
4 The evolution operator and its summability improving properties
Suppose that, besides Hypotheses 2.1, the assumptions on ψ in Theorem 3.6 hold true for any
4.1 Continuity properties of the nonlinear evolution operator
In the following theorem, assuming the above conditions on ψ, we prove an interesting continuity property of the operator
Theorem 4.1.
Let
Proof.
Let
for any
for any
and
in
for any
for any
for any
4.2 Hypercontractivity
Throughout this and the forthcoming subsections we set
for any smooth enough function ζ. To begin with, we recall the following crucial result.
Lemma 4.2 ([4, Lemma 3.1]).
Assume that Hypotheses 2.1 hold true and fix
Hypotheses 4.3.
We introduce the following assumptions.
There exists a nonnegative function
There exist locally bounded functions
for any
There exists a positive constant K such that
(4.3)for any
Remark 4.4.
(i)
Hypothesis 4.3 (i) implies that
(ii)
Sufficient conditions for (4.3) to hold are given in [4]. In particular, (4.3) holds true when (2.3) is satisfied with
We can now prove the main result of this subsection.
Theorem 4.5.
Let Hypotheses 2.1 and 4.3 be satisfied. Then,
for any
where
Proof.
To begin with, we observe that it suffices to prove (4.4) and (4.5)
for functions
We split the rest of the proof into two steps. In the first one we prove (4.4) and in the latter one (4.5).
Step 1. Fix
for any
for any
By definition, the function
Taking into account that
and
we deduce
Using Hypotheses 4.3 (i), (iv), the expression of the function
Further, since
it follows that
for any
for any
Hence,
for some continuous function
for some continuous function
for some positive and continuous function
where, again,
Hence, integrating (4.8) between s and t and letting first
Applying the Gronwall lemma, we conclude the proof of (4.4).
Step 2. To check estimate (4.5),
we arbitrarily fix
With these choices of ε and
From Step 1, we know that
By the evolution law and estimates (4.10) and (4.1) we get
for any
Replacing the value of ε in the expression of
and, since the function
4.3 Supercontractivity
In the next theorem we prove a stronger result than Theorem 4.5, i.e., we
prove that the nonlinear evolution operator
Hypothesis 4.7.
There exists a decreasing function
for any
Remark 4.8.
Sufficient conditions for (4.12) to hold are given in [3]. In particular, it holds true when (2.3) is satisfied with
Theorem 4.9.
Let Hypotheses 2.1, 4.3
(i)–(iii) and 4.7 be satisfied.
Then, for any
Here,
Proof.
The proof of this result follows the same lines of the proof of Theorem 4.5. For this reason we use the notation therein introduced and we limit ourselves to sketching it in the case when
Step 1. Here, we prove (4.13). For any
The function
and the same arguments used to prove (4.7) show that, if
Choosing
where
which yields
Now, for any
Step 2. Fix
The same arguments used in Step 2 of the proof of Theorem 4.5 show that
Estimate (4.14) follows with
4.4 Ultraboundedness
To begin with, we prove a sort of Harnack inequality, which besides the interest in its own will be crucial to prove the ultraboundedness of the nonlinear evolution operator
Proposition 4.10.
Let Hypotheses 2.1
(i)–(iii) and 4.3
(i)–(iii) be satisfied. Further, suppose that estimate (2.3) holds, with
where
Proof.
To begin with, we observe that it suffices to prove (4.16) for
functions in
So, let us fix
We observe that
and
Hence, we get
where
From Hypothesis 4.3 (i) it follows that
where
Moreover, using formula (2.2), we can estimate
These two estimates replaced in (4.17) give
Straightforward computations show that
where
Moreover, using the inequality
where
or even
By formula (2.2) we can let ε and δ tend to zero on both sides of the previous inequality and this yields the assertion. ∎
We can now prove the main result of this subsection. For this purpose, we
set
Hypothesis 4.11.
For any
Remark 4.12.
A sufficient condition for Hypothesis 4.11 to hold is given in [3, Theorem 4.3]. More precisely, it holds when (2.3) holds with
Theorem 4.13.
Assume that Hypothesis 4.11 and the conditions in Proposition 4.10 are
satisfied. Then, for any
for some continuous functions
Proof.
As usually, we prove the assertion for functions in
Step 1. We prove (4.21).
So, let us fix
where
where
Now, using the evolution law and again (4.16), we can write
for any
From (4.24), (4.25), choosing
and (4.21) follows with
Step 2. We fix
Hence, from (4.1) we get
Taking
5 Stability of the null solution
In this section we study the stability of the null solution to problem (1.2) both
in the
Theorem 5.1.
The following properties are satisfied.
Let Hypotheses 2.1, 4.3 (i)–(iii) hold true. Further, suppose that the constant
(5.1)
Proof.
(i) Estimate (5.1) can be obtained
arguing as in the proof of Theorem 4.5, where now
Hence, from (4.1) we obtain
where
(ii) The assertion follows easily letting p tend to
6 Examples
Here, we exhibit some classes of nonautonomous elliptic operators and some classes of nonlinear functions ψ which satisfy the assumptions of this paper.
Throughout this section, we fix a right-halfline I and assume that
for any
Example 6.1 (Local existence and regularity).
Fix
Here,
for any
for some positive constant C, independent of t, x,
If, in addition, g is γ- Hölder continuous for some
Example 6.2 (Global existence and stability).
Fix
where
If h is globally Lipschitz continuous, then ψ satisfies Hypotheses 4.3 (i) with
For any
then the functions
Example 6.3 (Summability improving properties).
We take the same function ψ as in Example 6.2. In view of Remark 4.4, we assume that the diffusion coefficients are independent of x.
Since
If, in addition, the power r in the drift coefficients is positive, then (4.12) holds true, by Remark 4.12.
Indeed, in such a case we can estimate
Funding statement: The authors are members of GNAMPA of the Italian Istituto Nazionale di Alta Matematica. This work has been supported by the INdAM-GNAMPA Project 2016 “Equazioni e sistemi di equazioni ellittiche e paraboliche associate ad operatori con coefficienti illimitati e discontinui”.
A Technical results
Proposition A.1.
Let Hypotheses 2.1 hold and let
for some
belongs to
and
for any
Finally, if
and
where
Proof.
Throughout the proof, we will make use of [5, Proposition 2.7], where it has been shown that, for any
To begin with, we observe that, for any
for any
If the function g is as in the statement of the proposition, then we can approximate it by a sequence
Using again (1.3), we obtain
Proving that z is continuous in
Fix
To conclude that
with
for any
Let us now assume that
with
Finally, to prove the differentiability of z, we introduce the sequence
As it is immediately seen,
for such values of
Lemma A.2.
Let J be an interval and let
Proof.
To begin with, we observe that for any
Finally, we prove some interior
Proposition A.3.
Let
Proof.
Throughout the proof, we denote by c a positive constant, independent of n and u, which may vary from line to line.
Let us fix
Since the function
we can write
where
for any
Now, for any
By interpolation and using Young’s inequalities we obtain, for any
and
Collecting the above estimates together, we get
Now we fix
Since
for any
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© 2019 Walter de Gruyter GmbH, Berlin/Boston
This work is licensed under the Creative Commons Attribution 4.0 Public License.
Articles in the same Issue
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- Asymptotic behavior of evolution systems in arbitrary Banach spaces using general almost periodic splittings
- Solvability of a product-type system of difference equations with six parameters
- On Dirichlet problem for fractional p-Laplacian with singular non-linearity
- Absence of Lavrentiev gap for non-autonomous functionals with (p,q)-growth
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- Hypercontractivity, supercontractivity, ultraboundedness and stability in semilinear problems
- Theoretical analysis of a water wave model with a nonlocal viscous dispersive term using the diffusive approach
- A multiplicity result for asymptotically linear Kirchhoff equations
- Higher-order anisotropic models in phase separation
- Well-posedness and maximum principles for lattice reaction-diffusion equations
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- Homoclinics for strongly indefinite almost periodic second order Hamiltonian systems
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Articles in the same Issue
- Frontmatter
- Asymptotic behavior of evolution systems in arbitrary Banach spaces using general almost periodic splittings
- Solvability of a product-type system of difference equations with six parameters
- On Dirichlet problem for fractional p-Laplacian with singular non-linearity
- Absence of Lavrentiev gap for non-autonomous functionals with (p,q)-growth
- On a class of fully nonlinear parabolic equations
- On sign-changing solutions for (p,q)-Laplace equations with two parameters
- Weighted Caffarelli–Kohn–Nirenberg type inequalities related to Grushin type operators
- On the fractional p-Laplacian equations with weight and general datum
- An elliptic equation with an indefinite sublinear boundary condition
- Liouville-type theorems for elliptic equations in half-space with mixed boundary value conditions
- Well/ill-posedness for the dissipative Navier–Stokes system in generalized Carleson measure spaces
- Hypercontractivity, supercontractivity, ultraboundedness and stability in semilinear problems
- Theoretical analysis of a water wave model with a nonlocal viscous dispersive term using the diffusive approach
- A multiplicity result for asymptotically linear Kirchhoff equations
- Higher-order anisotropic models in phase separation
- Well-posedness and maximum principles for lattice reaction-diffusion equations
- Existence of a bound state solution for quasilinear Schrödinger equations
- Existence and concentration behavior of solutions for a class of quasilinear elliptic equations with critical growth
- Homoclinics for strongly indefinite almost periodic second order Hamiltonian systems
- A new method for converting boundary value problems for impulsive fractional differential equations to integral equations and its applications
- Diffusive logistic equations with harvesting and heterogeneity under strong growth rate
- On viscosity and weak solutions for non-homogeneous p-Laplace equations
- Periodic impulsive fractional differential equations
- A result of uniqueness of solutions of the Shigesada–Kawasaki–Teramoto equations
-
Solutions of vectorial Hamilton–Jacobi equations are rank-one absolute minimisers in
- Large solutions to non-divergence structure semilinear elliptic equations with inhomogeneous term
- The elliptic sinh-Gordon equation in a semi-strip
- The Gelfand problem for the 1-homogeneous p-Laplacian
- Boundary layers to a singularly perturbed Klein–Gordon–Maxwell–Proca system on a compact Riemannian manifold with boundary
- Subharmonic solutions of Hamiltonian systems displaying some kind of sublinear growth
- Multiple solutions for an elliptic system with indefinite Robin boundary conditions
- New solutions for critical Neumann problems in ℝ2
- A fractional Kirchhoff problem involving a singular term and a critical nonlinearity
- Existence and non-existence of solutions to a Hamiltonian strongly degenerate elliptic system
- Characterizing the strange term in critical size homogenization: Quasilinear equations with a general microscopic boundary condition
- Nonlocal perturbations of the fractional Choquard equation
- A pathological example in nonlinear spectral theory
- Infinitely many solutions for cubic nonlinear Schrödinger equations in dimension four
- On Cauchy–Liouville-type theorems
- Maximal Lp -Lq regularity to the Stokes problem with Navier boundary conditions
- Besov regularity for solutions of p-harmonic equations
- The classical theory of calculus of variations for generalized functions
- On the Cauchy problem of a degenerate parabolic-hyperbolic PDE with Lévy noise
- Hölder gradient estimates for a class of singular or degenerate parabolic equations
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Critical and subcritical fractional Trudinger–Moser-type inequalities on
- Multiple nonradial solutions for a nonlinear elliptic problem with singular and decaying radial potential
- Quantization of energy and weakly turbulent profiles of solutions to some damped second-order evolution equations
- An elliptic system with logarithmic nonlinearity
- The Caccioppoli ultrafunctions
- Equilibrium of a production economy with non-compact attainable allocations set
- Exact behavior around isolated singularity for semilinear elliptic equations with a log-type nonlinearity
- The higher integrability of weak solutions of porous medium systems
- Classification of stable solutions for boundary value problems with nonlinear boundary conditions on Riemannian manifolds with nonnegative Ricci curvature
- Regularity results for p-Laplacians in pre-fractal domains
- Carleman estimates and null controllability of a class of singular parabolic equations
- Limit profiles and uniqueness of ground states to the nonlinear Choquard equations
- On a measure of noncompactness in the space of regulated functions and its applications
- p-fractional Hardy–Schrödinger–Kirchhoff systems with critical nonlinearities
- On the well-posedness of a multiscale mathematical model for Lithium-ion batteries
- Global existence of a radiative Euler system coupled to an electromagnetic field
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On the existence of a weak solution for some singular
- Choquard-type equations with Hardy–Littlewood–Sobolev upper-critical growth
- Clustered solutions for supercritical elliptic equations on Riemannian manifolds
- Ground state solutions for the Hénon prescribed mean curvature equation
- Quasilinear equations with indefinite nonlinearity
- Concentrating solutions for a planar elliptic problem with large nonlinear exponent and Robin boundary condition
- Retraction of: Concentrating solutions for a planar elliptic problem with large nonlinear exponent and Robin boundary condition
