Asymptotic behavior of evolution systems in arbitrary Banach spaces using general almost periodic splittings
-
Josef Kreulich
Abstract
We present sufficient conditions on the existence of solutions, with various specific almost periodicity properties, in the context of nonlinear, generally multivalued, non-autonomous initial value differential equations,
and their whole line analogues,
1 Introduction and preliminaries
We consider general closed and translation invariant subspaces of
We answer the question of when a solution of the nonlinear evolution equation
with
where
The underlying study is divided into sections where the nonlinearity is studied, which is even new in finite dimensions, and the appendices which show how Eberlein weak almost periodicity (Appendix A) and asymptotic almost periodicity (Appendix B) come into play, or give a hint on how continuous almost automorphy (Appendix C) becomes applicable.
In Appendix A we provide the analysis around Eberlein weakly almost periodic functions (EWAP).
We show, how Appendices A and B apply to obtain the splittings in Section 4.
We give certain conditions on
where A denotes a special linear operator and
Applying general existence results, even in the finite dimensional case, more general results on sums of dissipative operators are obtained while dispensing with the relationship between the ω-dissipativeness and the Lipschitz continuity of the nonlinear perturbation
Given a Banach space X, a function
is relatively compact with respect to the weak topology of the sup-normed
Banach space
Results of DeLeeuw and Glicksberg [6, 5] imply the following decomposition:
Here,
An exposition of this result is given in the book of Krengel [17]. For technical reasons we introduce the following spaces on the real line:
and
The decomposition of
and by the uniform continuity of Eberlein weakly almost periodic functions,
we obtain that for a given
2 Preliminaries on integral solutions
In [23] the following two types of equations have been discussed. The initial value problem
and the whole line equation on
with
Let X be a general Banach space.
As the given
Assumption 2.1.
The set
In comparison to the assumption given in [15], the uniform continuity on
Assumption 2.2.
There exist
for all
Recalling the study [23], the next assumption is stronger than that in [15] due to the Lipschitz continuity and linearized stability in
Assumption 2.3.
There exist bounded and Lipschitz continuous functions
for all
Remark 2.4.
Due to Assumptions 2.2 and 2.3, from [23, Definition 2.6, Remark 2.8], we read,
Due to the fact that we consider
and
Now we are in the situation to define the integral solution to the initial value problem. As the integral solution is defined with the help of Assumptions 2.2 and 2.3, it looks slightly different from the one in [15, Definition 6.18, pp. 217–218], which is due to Benilan.
Definition 2.5.
Let
for all
In order to solve the initial value problem (2.1), we use the Yosida approximation of the derivative. This leads to the equation
with the Yosida approximation
Thus, we obtain the solution
with (see [23, Remark 2.7])
The results for these approximations on
Theorem 2.6 ([23]).
Let
Further, we define an integral solution for the whole line problem (2.2).
Definition 2.7.
Let
for all
Similar to the initial value case, the Yosida approximation of the derivative was considered and led to the following result:
with
Again the
For these approximants we have the following result.
Theorem 2.8 ([23]).
Let
Let Assumptions 2.1 and 2.3 be fulfilled. Further, assume that the Lipschitz constant
Remark 2.9 ([22]).
The construction of the solutions implies
Moreover, we have a comparison result between the solution found on
Corollary 2.10 ([23]).
Let
3 Main definitions and context
We start with the notion of pseudo-resolvents:
Definition 3.1 ([25, Definition 7.1, p. 193]).
The members of a family of contractions,
are called pseudo-resolvents if
and
Lemma 3.2 ([25, Lemma 7.1, pp. 193–194]).
Pseudo-resolvents have the generator
Next we motivate a generalization for various types of almost periodicity.
Lemma 3.3.
Let Y be a closed linear subspace of
Proof.
For given
Then, for given
Consequently, for
and
Thus, we found that for
Due to the previous results it becomes straightforward to define types of almost periodicity even for multivalued ω-dissipative operators.
Definition 3.4.
Let Y be a closed linear translation-invariant subspace of
Remark 3.5.
The examples for Y are
This makes sense even for the splitting, as the almost periodic part is uniquely defined on
The control functions are extended in the same manner.
Due to Lemma 3.3, we can define the almost periodic part of a generally multivalued operator
with a projection
Definition 3.6.
A linear, closed and translation invariant subspace
and, finally,
Lemma 3.7.
Let Y split almost periodic, and let
with
then the members of the following family of operators are pseudo-resolvents:
If
Proof.
First we prove that
Next the range condition, but this is fulfilled, because we assumed
As
Using,
Applying (3.1), and evaluating at
Definition 3.8.
Let
Remark 3.9.
By the previous definition, for m-dissipative operators
where
Proof.
Using the fact that
and
The first equality comes with the dissipativeness of
Remark 3.10.
Let
Proof.
Given
Thus,
For an example, we recall the definition of demi-closedness [15, Definition 1.15, p. 18]
Definition 3.11.
An operator
Example 3.12.
Let
Let
Proof.
For a given
which is Lipschitz continuous, by the resolvent equation
and
For a given
To this end, we show that any subsequence of
The Lipschitz continuity of
To compute the almost periodic part of
To have a comparison between the classical almost periodicity and the generalized one, we provide the following proposition.
It shows that under certain conditions
Proposition 3.13.
Let
and assume that
Then, for
Proof.
To prove the equality we show that the resolvents coincide, i.e.,
The above together with Remark 3.9 will finish the proof. To prove (3.4) we need a representation of the resolvents. In doing this, we consider
For given
The Lipschitz continuity of
Consequently, for
Next we consider the resolvent of
Using the fact that B is Lipschitz for
Using the representation, we compute the almost periodic part and apply assumption (3.3), to obtain
We found that
which finishes the proof. ∎
Kenmochi and Otani [16] considered nonlinear evolution equations governed by time dependent subdifferential operators in Hilbert space H:
They constructed a not necessarily complete metric space of convex functions
Proposition 3.14.
Let
is almost periodic, then
is almost periodic for all
Proof.
We assume that for
The non-Cauchy assumption on
Note that
Clearly,
uniformly for
in the sense of Mosco, which implies, by using [1, Theorem 3.26, p. 305],
when
4 Main results
In this section we show how the previous results apply to evolution equations of the following type:
and the corresponding initial value problem
where
Definition 4.1.
A solution u of (4.2) is called asymptotically Y if there exists
The difference to earlier splitting results is that a splitting of the solution is found, but in the case of general dissipative and time dependent operators, the equation fulfilled by the almost periodic part was unknown.
Theorem 4.2.
Let, for a given
Let Y and
Proof.
The first part is a direct consequence of [22, Theorem 4.10]. For the second part, note that
and due to the assumption of an almost periodic splitting, we apply
Thus,
Applying Theorem 2.8,
Remark 4.3.
The notion of an integral solution in the case of a general splitting is not possible due to the missing control functions for
Proof.
Equation (4.5) and the fixpoint equation (4.4) show that we are in the situation of approximations used in the study [23, equation (31), p. 1076]. The uniqueness of fixpoints and the corresponding uniform limits (cf. Theorem 2.8) conclude the proof. ∎
Proposition 4.4.
In the context of Theorem 4.2, let
Proof.
Clearly,
pointwise in the norm of X.
As the solution on
Therefore, for
Using the fact that
Consequently, we may pass to the limit superior on both sides of (4.6), and by Fatou’s lemma we obtain the inequality. ∎
Corollary 4.5.
In the context of Theorem 4.2, let
If the control functions h, g belong to
Proof.
Apply inequality (4.6),
the existence of a sequence
Theorem 4.6.
Let
and
If the families
implies that
is asymptotically
Proof.
It is sufficient to prove that the almost periodic parts of
Thus, using the fixpoint equations for the approximants
Thus,
Corollary 4.7 ([22]).
Let
and let
and
Let also the families
then, for the corresponding evolution systems
i.e., they are asymptotically equivalent.
In the case of Eberlein weak almost periodicity, we obtain the following corollary.
Corollary 4.8.
Let
and
Let also the families
for all λ small,
for all
In the case of ordinary differential equations the result of Theorem 4.6 extends as follows.
Proposition 4.9.
Let
Further, let
and
Then
if and only if
Proof.
Due (4.8), the almost periodic parts of
which gives
5 Perturbations
In this section we put some general perturbation theorems for dissipative operators into the context of almost periodic splittings.
This extends the theory of semilinear operators like
Theorem 5.1.
Let
Proof.
We start with proving the ω-dissipativeness.
Let
Thus,
Consequently, we obtain that
To prove the m-dissipativeness and to obtain a representation of the resolvent of
with
An estimation gives that
We have
As the constants are contained in
From the previous step, we have
Noting that
Let
As a consequence, we obtain the same contraction mapping, and the uniqueness of the fixpoint concludes the proof. ∎
Remark 5.2.
In many studies the existence of solutions to equations like
need the precondition
If
is single-valued and, in case of a uniform convex dual, uniformly continuous on bounded subsets of X, cf. [15, Proposition 1.1, p. 2].
Note that a dissipative operator, in this case, is always strictly dissipative (i.e.,
Lemma 5.3.
Let
then, for
Further, if for
Additionally,
Proof.
As
Using Banach’s fixpoint theorem, we will find
Using
Thus,
Using
In sum this leads with the boundedness of
Recalling that
For a given
Thus, by the demiclosedness of
weakly, with
weakly in X, which concludes the proof. ∎
Theorem 5.4.
Let
Further, let
Proof.
The proof is analogous to that of [15, Corollary 1.25], proving the boundedness of
Remark 5.5.
To study the resolvents
Considering,
we find that the solutions are fixpoints of the contractions
Moreover, applying the bounded selection
then, by an observation similar to (5.7), we have
The uniform convexity of
which gives
Theorem 5.6.
Let
Proof.
We have to show that the almost periodic part of
From the Remark 5.5, for a given
As
Thus, the projection gives the fixpoint
Hence, we are in the situation of Remark 5.5, and can redo the final steps, similar to the proof of Lemma 5.3.
Consequently, for a given
weakly when
In the next example we show how the previous results apply to perturbations to the linear Dirichlet problem
Example 5.7.
Let
with ω its dissipativity constant,
and let
we have that
then we have
where
with
Proof.
From [15, Example 1.60 and Lemma 1.61, pp. 53–54], we obtain that
To view the case of Eberlein weak almost periodicity we have the following.
Example 5.8.
Let
Then
Proof.
Following the Definition A.4 and Theorem A.10, for given
The Eberlein weak almost periodicity is a consequence of the proof of Example 3.12, and the relative compact range comes with the compactness of
it remains to prove that
is continuous, which comes with the inequality
From Lemma 5.3 and Remark 5.5, we obtain the resolvent
uniformly for
Remark 5.9.
From Appendices B and C, we can obtain similar results for asymptotically almost periodic or continuous almost automorphic functions.
From Remark 3.9 and the previous perturbation results, we obtain the following corollary.
Corollary 5.10.
Let
and
If the families
Proof.
From the assumptions of Theorem 5.6 we obtain the m-dissipativeness for every operator
Finally, an application of Theorem 4.6 concludes the proof. ∎
A Eberlein weak almost periodicity
This and the remaining appendices discuss a number of closed and translation invariant subspaces of
Proposition A.1.
If
whenever the iterated limits exist, see [ 27 ].
If
exists uniformly over
Proposition A.2.
Let
Proof.
Let
The proposition above allows to assume, without loss of generality, that the limits in Proposition A.1 exist. Recall that only the equality of the limits has to be proven.
In this section we provide an extension of Traple’s result to infinite dimensions.
Theorem A.3 ([29]).
If X has finite dimension,
is continuous for every compact
Traple’s proof does not extend to the infinite dimensional case, for he used both
the algebraic structure of
In order to give sufficient conditions for the extension of Traple’s result,
hence existence of Eberlein weakly almost periodic solutions
to nonlinear differential equations in infinite dimensions,
we need the following definition, where
Definition A.4.
A function
for every compact and metric subset
We denote these functions by
With the aim to find a representation for the almost periodic part, we recall some technical results.
Proposition A.5 ([28]).
Eberlein weakly almost periodic functions are uniformly continuous.
Thus,
Remark A.6.
Let
Proof.
As K is compact metric, for given
There exists
From Definition A.4 (i), we obtain some
First we prove the boundedness.
For
To prove the uniform continuity, let
which finishes the proof. ∎
Remark A.7.
From the previous remark we conclude that if
Note that the continuity of ι does not imply a norm compact range for every
fulfills the previous definition.
Consequently, for each
will satisfy the assumptions of Definition A.4.
An example for
The restriction on K to be compact metric comes with the range of a function
Theorem A.8.
If
For the proof, the following technical lemma is needed.
Lemma A.9.
Let
be continuous.
Further, let a given sequence
Then the interchanged limits are equal for
and the iterated limits exist.
Proof.
Applying Proposition A.2 to bounded set on
By our hypothesis, we have
Now,
As the double limits on the right-hand side exist and are equal to 0, we proved
Proof of Theorem A.8.
To verify the double limits condition, we apply the previous lemma for given
to
From [8, Section V.6.3, p. 434], we recall that the weak topology on weakly compact subsets in separable B-spaces is a metric topology.
Noting that continuous images of separable spaces are separable,
we obtain that
Thus, we are in the situation of the previous lemma and our claim is proved. ∎
From the proof of Lemma A.9, and using that only local continuity is needed, we give the corollary for
which was introduced by Goldberg and Irwin [12].
Corollary A.10.
Let, for a Banach space Y,
Proof.
The reader will have no difficulty to apply the previous theorem to
For the second part, it remains to prove the compactness of
and the proof is complete. ∎
In [18, Example 2.17, p. 17], it is shown that the compactness assumption on the range of g is essential.
Corollary A.11.
Let
Proof.
Letting
the previous corollary serves for the proof. ∎
From Corollary A.10, we also obtain the next result of Goldberg and Irwin [12].
Corollary A.12.
If
Theorem A.8 gives a condition on f such that
it is also of interest when
where
Proposition A.13.
For the decomposition
we have that the projection
Proof.
For
which leads to the claim. ∎
Corollary A.14.
Any two functions
Proof.
First we consider the case where the almost periodic parts of f
and g are equal to zero, and let
By the proposition above, we have that every Eberlein weakly almost periodic function is uniformly continuous, hence the semigroup of translations
Thus, if
Thus, the desired result is a consequence of the classical diagonal process for the double sequence
If
we find, by the double limits criterion and the representation for the dual of
whenever the iterated limits exist. However, by a successive diagonalisation of the double sequences
we may assume that their individual iterated limits exist and are equal, hence h is Eberlein weakly almost periodic.
Thus, h has a unique decomposition into an almost periodic and a
Clearly, from the decomposition of f and g, we obtain
Further, by the observation in the first part of this proof,
we have that
is the desired one. ∎
Lemma A.15.
For every
for every
Proof.
Given any
Since
is continuous, and
Consequently, L is a subset of a closed and separable subspace Y of
By the fact that
we obtain the relative weak compactness for L. Hence, the weak topology on L is metrizable, and we may choose a metric of the form
Choosing
we construct a dense sequence
converges weakly as
for all
Now, for a given
where the last inequality follows from the definition of the metric and the fact that the norm of the projection on the almost periodic part is less or equal to one. This completes the proof. ∎
Theorem A.16.
Let
where
Proof.
From the notation of the theorem, we find that
By Theorem A.8, F fulfills the hypothesis of Lemma A.15.
Note that F is a function on discrete points
Using that all the sequences are convergent, it remains to compute the limit, which can be done in the pointwise weak topology as follows:
Since
Corollary A.17.
If for a Banach space Y,
Proof.
Using that on
Remark A.18.
We consider the context of Example 5.7 with
and
Then, for
Proof.
We give a proof for
which leads to a single sequence
weakly in
and
which concludes the proof. ∎
Remark A.19.
Note that the methods apply in a similar way, when
B Asymptotically almost periodic functions
The classical concepts are due to Frechet [11, 10]. By [11, 10, 6, 5], we have
In consequence, we have the following proposition.
Proposition B.1.
For the decomposition
we have that the projection
Using compactness methods we have the following theorem.
Theorem B.2.
Let
Proof.
As
C Almost automorphic functions
Bochner introduced the notion of almost automorphy.
Definition C.1.
A function
and
We define
If the limit g is continuous, then f is called continuous (Bochner)-almost automorphic. We define
Noting, that for
The following Theorem is due to [30, Lemma 4.1.1, p. 742].
Theorem C.2.
Continuous almost automorphic functions are uniformly continuous, i.e.,
Remark C.3.
In [7], the asymptotically almost automorphic functions
For suitable
was computed.
Thus, the underlying study becomes applicable, when switching from almost automorphy to continuous almost automorphy, and adding for
and the projection has a norm less than one.
References
[1] H. Attouch, Variational Convergence for Functions and Operators, Pitman, Boston, 1984. Suche in Google Scholar
[2] B. Aulbach and N. Van Minh, A sufficient condition for almost periodicity of nonautonomous nonlinear evolution equations, Nonlinear Anal. 51 (2002), 145–153. 10.1016/S0362-546X(01)00816-1Suche in Google Scholar
[3] P. Cieuta, S. Fatajou and G. N’Guérékata, Bounded and almost automorphic solutions of some semilinear differential equations in Banach spaces, Nonlinear Anal. 71 (2009), 74–684. 10.1016/j.na.2008.10.100Suche in Google Scholar
[4] M. G. Crandall and A. Pazy, Nonlinear evolution equations in Banach spaces, Israel J. Math. 11 (1972), 57–94. 10.1007/BF02761448Suche in Google Scholar
[5] K. de Leeuw and I. Glicksberg, Almost periodic functions on semigroups, Acta Math. 105 (1961), 99–140. 10.1007/BF02559536Suche in Google Scholar
[6] K. de Leeuw and I. Glicksberg, Applications of almost periodic compactifications, Acta Math. 105 (1961), 63–97. 10.1007/BF02559535Suche in Google Scholar
[7] H. S. Ding, J. Liang and T. J. Xiao, Asymptotically almost automorphic solutions for some integrodifferential equations with nonlocal initial conditions, J. Math Anal. Appl. 338 (2008), 1188–1199. 10.1016/j.jmaa.2007.05.014Suche in Google Scholar
[8] N. Dunford and J. T. Schwartz, Linear Operators. Part I, John Wiley & Sons, New York, 1957. Suche in Google Scholar
[9] W. F. Eberlein, Abstract ergodic theorems and weak almost periodic functions, Trans. Amer. Math. Soc. 67 (1949), 217–240. 10.1090/S0002-9947-1949-0036455-9Suche in Google Scholar
[10] M. Frechet, Les fonctions asymptotiquement presque-périodiques, Rev. Sci 79 (1941), 341–354. Suche in Google Scholar
[11] M. Frechet, Les fonctions asymptotiquement presque-périodiques continues, C. R. Acad. Sci. Paris 213 (1941), 520–522. Suche in Google Scholar
[12] S. Goldberg and P. Irwin, Weakly almost periodic vector valued functions, Diss. Math. 157 (1979), 1–42. Suche in Google Scholar
[13] A. Haraux, Asymptotic behavior of trajectories for some nonautonomous, almost periodic processes, J. Differential Equations 49 (1983), 473–483. 10.1016/0022-0396(83)90008-6Suche in Google Scholar
[14] Y. Hino, T. Naito, N. V. Minh and J. S. Shin, Almost Periodic Solutions of Differential Equations in Banach Spaces, Taylor & Francis, London, 2002. 10.1201/b16833Suche in Google Scholar
[15] K. Ito and F. Kappel, Evolution Equations and Approximations, Ser. Adv. Math. Appl. Sci. 61, World Scientific, Singapore, 2002. 10.1142/4990Suche in Google Scholar
[16] N. Kenmochi and M. Otani, Nonlinear evolution equations governed by subdifferential operators with almost periodic time dependence, Rend. Accad. Naz. Sci XL V. Ser. Mem. Mat. 10 (1986), 65–91. Suche in Google Scholar
[17] U. Krengel, Ergodic Theorems, De Gruyter Stud. Math. 6, De Gruyter, Berlin, 1985. 10.1515/9783110844641Suche in Google Scholar
[18] J. Kreulich, Eberlein weak almost periodicity and differential equations in Banach spaces, Ph.D. thesis, Universität Essen, 1992. Suche in Google Scholar
[19] J. Kreulich, The RAGE theorem in Banach spaces, Semigroup Forum 49 (1994), 151–163. 10.1007/BF02573481Suche in Google Scholar
[20] J. Kreulich, Eberlein weakly almost periodic solutions of evolution equations in Banach spaces, Differential Integral Equations 9 (1996), 1005–1027. 10.57262/die/1367871528Suche in Google Scholar
[21] J. Kreulich, Bounded solutions of nonlinear Cauchy problems, Abstr. Appl. Anal. 7 (2002), no. 12, 637–661. 10.1155/S1085337502208015Suche in Google Scholar
[22] J. Kreulich, Asymptotic equivalence of nonlinear evolution equations in Banach spaces, J. Evol. Equ. 14 (2014), 969–1000. 10.1007/s00028-014-0248-0Suche in Google Scholar
[23] J. Kreulich, Asymptotic behaviour of nonlinear evolution equations in Banach spaces, J. Math. Anal. Appl. 424 (2015), no. 2, 1054–1102. 10.1016/j.jmaa.2014.10.028Suche in Google Scholar
[24] P. Milnes, On vector-valued weakly almost periodic functions, J. Lond. Math. Soc. (2) 22 (1980), 467–472. 10.1112/jlms/s2-22.3.467Suche in Google Scholar
[25] I. Miyadera, Nonlinear Semigroups, Transl. Math. Monogr. 109, American Mathematical Society, Providence, 1992. 10.1090/mmono/109Suche in Google Scholar
[26] J. Prüss, Bounded solutions of Volterra equations, SIAM J. Math. Anal. 19 (1988), 133–149. 10.1137/0519010Suche in Google Scholar
[27] W. M. Ruess and W. H. Summers, Integration of asymptotically almost periodic functions and weak asymptotic almost periodicity, Diss. Math. 279 (1989), 1–35. Suche in Google Scholar
[28] W. M. Ruess and W. H. Summers, Ergodic theorems for semigroups of operators, Proc. Amer. Math. Soc. 114 (1992), 423–432. 10.1090/S0002-9939-1992-1075950-1Suche in Google Scholar
[29] J. Traple, Weakly almost periodic solutions of differential equations, J. Differential Equations 45 (1982), 199–206. 10.1016/0022-0396(82)90066-3Suche in Google Scholar
[30] W. A. Veech, Automorphic functions on groups, Amer. J. Math. 87 (1965), no. 3, 719–751. 10.2307/2373071Suche in Google Scholar
[31] T-J. Xiao, J. Liang and J. Zhang, Pseudo almost automorphic solutions to semilinear differental equations in Banach spaces, Semigroup Forum 76 (2008), 518–524. 10.1007/s00233-007-9011-ySuche in Google Scholar
© 2019 Walter de Gruyter GmbH, Berlin/Boston
This work is licensed under the Creative Commons Attribution 4.0 Public License.
Artikel in diesem Heft
- 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
-
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
-
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
Artikel in diesem Heft
- 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
-
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
-
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
