This paper is devoted to continuing the analysis started in . We consider a family of linear second-order differential operators acting on smooth functions ζ as
where I is either an open right halfline or the whole . Then, given , we are interested in studying the nonlinear Cauchy problem
where . We assume that the coefficients and (), possibly unbounded, are smooth enough, the diffusion matrix is uniformly elliptic and there exists a Lyapunov function φ for (see Hypothesis 2.1 (iii)). These assumptions yield that the linear part generates a linear evolution operator in . More precisely, for every and , the function belongs to , it is the unique bounded classical solution of the Cauchy problem (1.2), with , and satisfies the estimate
We refer the reader to  for the construction of the evolution operator and for further details.
Classical arguments can be adapted to our case to prove the existence of a unique local mild solution of problem (1.2) for any , i.e., a function (for some ) such that
Under reasonable assumptions such a mild solution is classical, defined in the whole and satisfies the condition
for any . Hence, setting for any we deduce that maps into and, from the uniqueness of the solution to (1.2), it follows that it satisfies the evolution law
for any and .
As in the linear case we are also interested to set problem (1.2) in an -context. However, as it is already known from the linear case, the most natural -setting where problems with unbounded coefficients can be studied is that related to the so-called evolution systems of measures , that is one-parameter families of Borel probability measures such that
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 (see Section 2), there exists a unique tight evolution system of measures , which has the peculiarity to be the unique system related to the asymptotic behavior of as t tends to . We also mention that, typically, even if for , the measures and are equivalent (being equivalent to the restriction of the Lebesgue measure to the Borel σ-algebra in ), the corresponding -spaces differ.
Formula (1.5) and the density of in allow to extend to a contraction from to for any and any and to prove very nice properties of in these spaces.
In view of these facts, it is significant to extend to an operator from to for any . This can be done if (see Hypothesis 2.1 (v)), is Lipschitz continuous in uniformly with respect to and, in addition, . In particular, each operator is continuous from to .
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 . This can be easily seen in the particular case of the one-dimensional Ornstein–Uhlenbeck operator, where the evolution system of measures is replaced by a time-independent measure μ (the so-called invariant measure), which is the Gaussian centered at zero with covariance . For any , the function belongs to for any but it does not belong to .
Under the previous assumptions, for any , can be identified with the unique mild solution to problem (1.2) which belongs to , for any , such that for almost every . Here, μ is the unique Borel measure on the σ-algebra of all the Borel subsets of which extends the map defined on the product of a Borel set and a Borel set by
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 are not immediate and true in all the cases. For this reason in Section 4 we investigate properties such as hypercontractivity, supercontractivity, ultraboundedness of the evolution operator and its spatial gradient. Differently from , where and the hypercontractivity of is proved assuming for any , here we consider a more general case. More precisely we assume that there exist and such that for any , , . Under some other technical assumptions on the growth of the coefficients and as , we show that as in the linear case, (see [3, 4]), the hypercontractivity and the supercontractivity of and are related to some logarithmic Sobolev inequalities with respect to the tight system . These estimates are the natural counterpart of the Sobolev embedding theorems in the context of invariant measures and evolution systems of measures.
For what concerns the ultraboundedness of and we first prove an Harnack-type estimate which establishes a pointwise estimate of in terms of for any , and . This estimate, together with the evolution law and the ultraboundedness of , allow us to conclude that, for any and any , the function belongs to and to prove an estimate of in terms of .
Finally, assuming that for every and , we prove that the trivial solution to the Cauchy problem (1.2) is exponentially stable both in and in . This means that as for some constants and , both when and . In the first case, the space X depends itself on t. We stress that, under sufficient conditions on the coefficients of the operators , which include their convergence at infinity, in [2, 11] it has been proved that the measure weakly converges to a measure μ, which turns out the invariant measure of the operator , whose coefficients are the limit as of the coefficients of the operator . This gives more information on the convergence to zero of at infinity. We refer the reader also to  for the case of T-time periodic coefficients.
To get the exponential stability of the trivial solution in , differently from  where a nonautonomous version of the principle of linearized stability is used and more restrictive assumptions on ψ are required, we let p tend to in the decay estimate of , since all the constants appearing in this estimate admit finite limit as p tends to . In particular, we stress that we do not need any additional assumptions on the differentiability of ψ but, on the other hand, we require that the mild solution of (1.2) is actually classical.
For , by we mean the space of the functions in which are bounded together with all their derivatives up to the -th order. is endowed with the norm
where denotes the integer part of k. When , we use the subscript “loc” to denote the space of all such that the derivatives of order are )-Hölder continuous in any compact subset of . Given an interval J, we denote by and (), respectively, the set of all functions such that is Lipschitz continuous in , uniformly with respect to , and the usual parabolic Hölder space. The subscript “loc” has the same meaning as above.
We use the symbols , and to denote respectively the time derivative and the spatial derivatives and for any .
The open ball in centered at 0 with radius and its closure are denoted by and , respectively. For any measurable set A, contained in or in , we denote by the characteristic function of A. Finally, we write when A is compactly contained in B.
2 Assumptions and preliminary results
Let be the family of linear second-order differential operators defined by (1.1).
Our standing assumptions on the coefficients of the operators are as follows.
The coefficients belong to for any and some .
For every , the matrix is symmetric and there exists a function
with positive infimum , such that
for any and any .
There exists a nonnegative function , diverging to as , such that
for any and some positive constants a and c.
There exists a locally bounded function such that
for any and any , where κ is defined in (ii).
There exists a function such that
for any and . Further, there exists such that
Under Hypotheses 2.1 (i)–(iii) (actually even under weaker assumptions) it is possible to associate an evolution operator to the operator in , as described in the Introduction. The function is continuous in and
where are probability measures for any , . This implies that
for any , and . Moreover, Hypotheses 2.1 (iv) and (v) yield the pointwise gradient estimates
for any , , , , and some positive constants and , where is given by (2.1), with p instead of . We stress that the pointwise estimates (2.3) and (2.4) have been proved with the constants and also depending of p. Actually, these constants may be taken independent of p. Indeed, consider for instance estimate (2.4). If , then using the representation formula (2.2) we can estimate
for any and , and, hence, estimate (2.4) holds true with a constant which can be taken independent of p.
The case in estimate (2.3) is much more delicate and requires stronger assumptions. Indeed, as  shows, the algebraic condition in for any with is a necessary condition for (2.3) (with ) to hold. For this reason, if the diffusion coefficients are bounded and independent of x, then the pointwise gradient estimate (2.3) holds true also with and , where is the supremum over of the function r in Hypothesis 2.1 (v).
Under Hypotheses 2.1 we can also associate an evolution system of measures with the operators . Such a family of measures is tight, namely for every there exists such that for any . The invariance property (1.5) and the density of in , , allows to extend to a contraction from to for any . As it has been stressed in the Introduction, in general evolution systems of measures are infinitely many, but, under suitable assumptions, there exists a unique tight evolution system of measures. This is, for instance, the case when Hypotheses 2.1 are satisfied as well as the following two conditions:
and belong to for any and some . Moreover, belongs to and belong to for any , and some .
There exists a constant such that either and for any , or the diffusion coefficients are bounded in .
3 The semilinear problem in a bounded time interval
Given , we are interested in studying the Cauchy problem (1.2) both in the case when and in the case when .
Our standing assumptions on ψ are as follows.
The function is continuous. Moreover, there exists such that for any and some constant
for any , , , .
The function belongs to .
where is the constant in (2.4). Moreover, for any , and , belongs to and there exists a positive constant such that
Finally, if is such that , then
Even if the proof is quite standard, for the reader’s convenience we provide some details.
Fix . Let be such that , where and is the constant in (2.4). Further, for any , let be the set of all such that .
Step 1. We prove that there exists such that, for any satisfying the condition
there exists a mild solution to problem (1.2) defined in the time interval . For this purpose, we consider the operator Γ, defined by the right-hand side of (1.4) for any (the ball of centered at zero with radius ). Clearly, the function is continuous in and is bounded in for any . Moreover, estimating and taking (3.1) into account, we can easily show that the function is bounded in . Hence, Proposition A.1 and estimates (1.3) and (2.4) show that for any and . To show that, for a suitable , Γ is a -contraction in , we observe that, using again (3.1), it follows that
for any , where is the constant in Hypothesis 3.1 (i). From this inequality and estimates (1.3) and (2.4) we conclude that for any , where , as the forthcoming constants, is independent of δ and u, if not otherwise specified. Hence, choosing δ properly, we can make Γ a -contraction in .
It is also straightforward to see that Γ maps into itself, up to replacing δ with a smaller value if needed. It suffices to split , use the previous result and estimate
for , any , where . Estimating with for any , from (3.5), with , it follows that
The same arguments show that can be estimated pointwise in by the right-hand side of (3.7), with being possibly replaced by a larger constant . Summing up, we have proved that
The generalized Gronwall lemma (see ) yields for any , i.e., in .
Step 4. We prove that for any , , , and
for some constant , independent of f. For this purpose, we observe that the results in the previous steps show that the function satisfies the estimate for any , the constant being independent of f. Applying Proposition A.1 and estimate (A.5), we complete the proof. ∎
In addition to the assumption of Theorem 3.2 suppose that there exist and such that and
Fix . Theorem 3.2 shows that belongs to and
for any . Moreover, by interpolation from (3.2) it follows that for any . From these estimates, adding and subtracting , we deduce that for any , such that and some positive constant C, depending on R and u. As a byproduct, for any and some positive constant , depending on R and u. Now, using Proposition A.1, we conclude that for any if , and for otherwise. ∎
Suppose that (3.1) is replaced by the condition
in , for any , , and some positive constant . Then, the proof of the previous theorem can be repeated verbatim with , endowed with the natural norm, and we can show that the mild solution to problem (1.2) belongs to and for some positive constant , independent of f.
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).
We introduce the following assumptions.
For any there exists a positive constant such that
for any , , and .
For any there exist positive constants , and a, and a function with nonnegative values and blowing up at infinity such that and in for any , and .
In the rest of this section, for any and we denote by the supremum over of the function ; is defined similarly, replacing by .
Assume that Hypotheses 2.1, 3.1 (ii), 3.5 and condition (3.9) are satisfied. Then, for any , the classical solution to problem (1.2) exists in . If, further, the constant in Hypothesis 3.5 (i) is independent of R, then for any ,
for every , where for some positive constants and .
We split the proof into two steps.
Step 1. We prove that, for any , is defined in the whole . To this end, we fix , denote by the maximal time domain where is defined and assume, by contradiction, that . We are going to prove that is bounded in . Once this is proved, we can use Hypotheses 3.5 (i) to deduce, adding and subtracting , that for , and some constant , which depends on and . Applying the same arguments as in Step 2 of the proof of Theorem 3.2, we can show that also the function is bounded in . This is enough to infer that can be extended beyond , contradicting the maximality of the interval .
To prove that is bounded in , we fix , and we set
for any . A straightforward computation shows that
in . Since is bounded in and blows up at infinity, the function admits a maximum point . If for any n, then in . Assume that for some n. If , then . If , then , so that, multiplying both the sides of (3.12) by and using Hypotheses 3.5 (ii) we get , which clearly implies that, also in this case, u is bounded from above in by a constant, independent of b.
Repeating the same arguments with being replaced by , we conclude that is bounded also from below by a positive constant independent of b. Since b is arbitrary, it follows that as claimed.
Step 2. Fix , and let be the norm defined by
on smooth functions v, where ω is a positive constant to be chosen later on and to fix the ideas we assume that . From Hypothesis 3.5 (i), where is replaced by a constant L, it follows that
for any . Hence, recalling that each operator is a contraction from to and using the second pointwise gradient estimate in (2.4) and the invariance property of the family , we conclude that
To estimate the integral terms in the last side of (3.13), we fix and observe that
Hence, minimizing over , we conclude that the left-hand side of estimate (3.14) is bounded from above by . Splitting and arguing as above, also the last term in square brackets in the last side of (3.13) can be estimated by . It thus follows that
where . Choosing ω such that , we obtain
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 when , that is a function , for any , such that for almost every and, for such values of t, the equality
holds true in , where is negligible with respect to the measure (or, equivalently, with respect to the restriction of the Lebesgue measure to the Borel σ-algebra in ).
Under all the assumptions of Theorem 3.6, for any there exists a unique mild solution to the Cauchy problem (1.2). The function satisfies estimates (3.10) and (3.11) with the supremum being replaced by the essential supremum and, as a byproduct, if , and for any otherwise. Finally, if there exists such that
for any , , , , , and a constant , then, for any and almost every , belongs to . Moreover, and satisfies the equation .
Fix and let be a sequence converging to f in . By (3.11), is a Cauchy sequence in for any . Hence, there exists a function v such that converges to in for any . Moreover, writing (3.10), with f being replaced by , and letting n tend to we deduce that v satisfies (3.10) as well.
Next, using (3.11) we can estimate
for any if and for otherwise. Hence, recalling that for any , we conclude that the sequence converges in to a function, which we denote by . Clearly, almost everywhere in for almost every . Letting n tend to in formula (1.4), with replacing f, we deduce that is a mild solution to problem (1.2). The uniqueness follows, arguing as in the proof of Theorem 3.2 with the obvious changes.
Let us now prove the last part of the statement. We again use an approximation argument. Fix and . At a first step, we estimate the norm of the operator in and in , for any . In the rest of the proof, we denote by c a positive constant, possibly depending on R, but being independent of t, r and , which may vary from line to line. Since there exists a positive and continuous function such that , the spaces and coincide and their norms are equivalent for any . From this remark, the interior -estimates in Theorem A.3, with and the contractiveness of from to , imply that
first for any , and then, by density, for any . Since, for ,
with equivalence of the corresponding norms, by an interpolation argument and (3.16) we deduce that for any . Hence, if for any we consider the function , which is the integral term in (1.4), with u being replaced by , and use (3.11) and the fact that , then we get
for any . We have so proved that, for any and almost every , the function belongs to and
Using these estimates, we can now show that , for any . For this purpose, we add and subtract , use condition (3.15) and the Lipschitz continuity of ψ with respect to the last two variables to infer that
for any , and . Hence, using (3.17) we obtain
and, using (3.18),
From these two estimates we conclude that
for any , any and any . Hence, for any and β, such that , a long but straightforward computation reveals that
for any . We are almost done. Indeed, by interpolation from Proposition A.3 we deduce that . From this and the previous estimate we conclude that
for any and , so that
for any , thanks to (3.16). From this estimate it is easy to deduce that is a Cauchy sequence in . Since is a classical solution to problem (1.2), we conclude that is a Cauchy sequence in . It thus follows that and it solves the equation
in . ∎
Under Hypotheses 2.1, the following properties are satisfied.
Let and for some . Then, for any , the Cauchy problem ( 1.2 ) admits a unique mild solution which belongs to , if , and to for any , if . Further, satisfies ( 3.10 ) and ( 3.11 ), with the supremum being replaced by the essential supremum.
To prove property (i), it suffices to apply the Banach fixed point theorem in the space of all the functions such that , where is defined in Step 2 of the proof of Theorem 3.6, with . The uniqueness of the so obtained solution follows from the condition , in a standard way.
To prove property (ii), one can argue by approximation. We fix , approximate it by a sequence , converging to f in , and introducing a standard sequence of cut-off functions. If we set for any , then each function satisfies the assumptions in property (i) and . Therefore, the Cauchy problem (1.2), with and replacing f and ψ admits a unique mild solution , which satisfies (3.10) and (3.11) with replacing f. The arguments in the first part of the proof of Corollary 3.7 allow us to prove the existence of a mild solution to the Cauchy problem (1.2) with the properties in the statement of the proposition. The uniqueness of the solution follows also in this case from the condition . ∎
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 or for each and . Then, for any and the mild solution to problem (1.2) exists in the whole of . Hence, we can set for any . Each operator maps into . Moreover, the uniqueness of the solution to problem (1.2) yields the evolution law for any and . Hence is a nonlinear evolution operator in . It can be extended to the -setting, for any , using the same arguments as in the first part of the proof of Corollary 3.7. Clearly, if is Lipschitz continuous in , uniformly with respect to , for any , then by density, we still deduce that satisfies the evolution law and, moreover, each operator is bounded from to and
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 .
Let be a bounded sequence converging to some function pointwise in . Then, for any , and converge to and , respectively, locally uniformly in .
Let and f be as in the statement. To ease the notation, we write and for and , respectively. Moreover, we set for any , and , and we denote by any constant such that
in , for any and some positive constant . Hence, the function satisfies the differential inequality
for any and . Since is continuous in and for some positive constant , independent of n, and any , we can apply [8, Lemma 7.1] and conclude that
for any . Hence,
for any . By [9, Proposition 3.1 (i)], vanishes as for any . Hence, by dominated convergence, vanishes as for any , which means that, for any , and converge uniformly in to and , respectively. The arbitrariness of R and T yields the assertion. ∎
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 . If and is constant outside a compact set K for every , then the function is continuously differentiable in and
We introduce the following assumptions.
, condition (3.9) is satisfied in , for any and some constant which may depend also on s and T, and there exist two constants and such that for any , and .
There exists a nonnegative function , blowing up at infinity such that in for some locally bounded functions .
There exist locally bounded functions such that
for any and any .
There exists a positive constant K such that
for any , and .
(i) Hypothesis 4.3 (i) implies that is bounded in and
We can now prove the main result of this subsection.
To begin with, we observe that it suffices to prove (4.4) and (4.5) for functions . Indeed, in the general case, the assertion follows approximating f with a sequence which converges to f in . By (3.10), converges to in for almost every . Hence, writing (4.4) and (4.5) with f being replaced by and letting n tend to , the assertion follows at once by applying Fatou lemma.
Step 1. Fix , , and set
for any , where and for any and . Here, ζ is a smooth function such that . Moreover, we set
for any . We recall that in [9, Theorem 5.4] it has been proved that . Hence, the functions are bounded in I and pointwise converge to zero as .
Taking into account that
Further, since and
it follows that
for any and some continuous function . Moreover, applying Hölder and Young inequalities and Hypothesis 2.1 (ii) we can infer that
for any and
for some continuous function . Now, we estimate the integral term containing . We begin by observing that
for some continuous function . Moreover,
for some positive and continuous function . Hence, replacing these estimates in (4.6), we get
where, again, is a continuous function. Choosing , if , otherwise, and then small enough we obtain
Hence, integrating (4.8) between s and t and letting first and then , by dominated convergence we get
Applying the Gronwall lemma, we conclude the proof of (4.4).
Step 2. To check estimate (4.5), we arbitrarily fix , and we take
With these choices of ε and , we have
From Step 1, we know that and
for any . In particular, taking and using Remark 4.4 (i) to estimate , we get
Replacing the value of ε in the expression of (see (4.9)), we deduce that
and, since the function is decreasing, . Finally, observing that is bounded in , (which follows from (4.9) recalling that ) and as , formula (4.5) follows immediately replacing in (4.11) the value of ε given by (4.9). ∎
In the next theorem we prove a stronger result than Theorem 4.5, i.e., we prove that the nonlinear evolution operator satisfies a supercontractivity property. For this purpose, we introduce the following additional assumption.
There exists a decreasing function blowing up as σ tends to such that
for any , and .
Here, are continuous functions such that .
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 and any , we set
and the same arguments used to prove (4.7) show that, if , then
Choosing , if , otherwise, and we get
where . Hence, integrating (4.15) between s and t and letting first and then , by dominated convergence we get
Now, for any and , we fix . We get and from the previous inequality the claim follows with
Step 2. Fix . By Step 1, belongs to and
The same arguments used in Step 2 of the proof of Theorem 4.5 show that for any and
Estimate (4.14) follows with , . ∎
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 .
where if and otherwise.
To begin with, we observe that it suffices to prove (4.16) for functions in . Indeed, if , we can determine a sequence , bounded with respect to the sup-norm and converging to f locally uniformly in . Writing (4.16) with f replaced by and using Theorem 4.1 and [9, Proposition 3.1 (i)], we can let n tend to and complete the proof.
So, let us fix and set for any and , where , (see Theorem 3.6), and is a standard sequence of cut-off functions. We note that for any and any . This is clear if . Suppose that . If , then is positive since . If , then which is positive if we choose large enough such that . Moreover, . Hence and we have
We observe that
Hence, we get
From Hypothesis 4.3 (i) it follows that
where . Using the Cauchy–Schwarz inequality, we can estimate
Moreover, using formula (2.2), we can estimate
These two estimates replaced in (4.17) give
Straightforward computations show that and vanish pointwise in as , for any and there exists a positive constant C such that in for any , thanks to Hypothesis 4.3 (iii). By [9, Lemma 3.4] the function is bounded in for any . Hence, by dominated convergence we conclude that vanishes as , pointwise in , for any and
where . Similarly, the last but one term in (4.18) vanishes pointwise in as , for any and
Moreover, using the inequality for any and , and that for any and any , we deduce
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 for any and , and introduce the following additional assumption.
For any and , the function belongs to and, for any , .
for some continuous functions which blow up at zero.
As usually, we prove the assertion for functions in .
where . By the tightness of the family we can fix such that for any and, from the previous chain of inequalities, we conclude that
Now, using the evolution law and again (4.16), we can write
and (4.21) follows with
Step 2. We fix , . By Theorem 3.2, and, by Step 1,
Hence, from (4.1) we get
Taking , estimate (4.22) follows with and . ∎
5 Stability of the null solution
In this section we study the stability of the null solution to problem (1.2) both in the - and -settings. For this reason, we assume that .
The following properties are satisfied.
Let Hypotheses 2.1, 4.3 (i)–(iii) hold true. Further, suppose that the constant is negative, where and are defined in Hypothesis 4.3 (ii). Then, for any , there exists a positive constant such that, for any , and ,
Suppose that the assumptions of Theorem 3.6 are satisfied. Further, assume that Hypotheses 4.3 (i)–(iii) hold with . Then ( 5.1 ) holds true for any with and and being replaced, respectively, by and .
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 is defined by (1.1) with
for any , where , the function is bounded with positive infimum and are the entries of a symmetric and positive definite matrix , for any , which satisfies the estimate for any , and some positive constant . Then Hypotheses 2.1 are satisfied with for any .
Example 6.1 (Local existence and regularity).
Fix and let us consider as nonlinear term the function defined by
Here, and are bounded and continuous functions with positive infimum,
for any and some positive constants and η, and . We claim that ψ satisfies Hypotheses 3.1 so that Theorem 3.2 can be applied to deduce that for any there exists a unique local mild solution to the Cauchy problem (1.2) with these choices of ψ and . Hypothesis 3.1 (ii) is trivially satisfied. To prove Hypothesis 3.1 (i), we fix , and . Then, for any and , we can estimate
for some positive constant C, independent of t, x, and (). Hence, Hypothesis 3.1 (i) is satisfied with and
Example 6.2 (Global existence and stability).
Fix and let be the function defined by
where has positive infimum and , for some , has positive infimum . The function is nonnegative, locally Lipschitz continuous in and satisfies the conditions and for some positive constants and . Finally, the function is Lipschitz continuous and vanishes at 0. As it is easily seen, ψ satisfies Hypotheses 3.5, condition (3.9) and, clearly, is a bounded function. Moreover, if we take for any , then also the condition in is satisfied with some suitable positive constants a and . Hence, by Theorem 3.6, problem (1.2) admits a unique global classical solution defined in the whole .
If h is globally Lipschitz continuous, then ψ satisfies Hypotheses 4.3 (i) with
then the functions and exponentially decay to zero as for any and , where is the constant introduced at the beginning of this section. The same result holds for any if we replace the -norm by the -norm.
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 for any and , and is positive, by [4, Theorem 3.3] Hypothesis 4.3 (iv) is satisfied. Then, by Theorem 4.5, estimates (4.4) and (4.5) are satisfied.
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 for any and , so that, by Theorem 4.13, the nonlinear evolution operator satisfies estimates (4.21) and (4.22).
A Technical results
Let Hypotheses 2.1 hold and let satisfy
for some and some . Then the function , defined by
belongs to for any ,
for any , and some positive constants and . In particular, if , then is bounded in .
Finally, if for some and any , then one has . Moreover,
where if , whereas ρ can be arbitrarily fixed in otherwise
Throughout the proof, we will make use of [5, Proposition 2.7], where it has been shown that, for any , , and , there exist positive constants and such that for any ,
To begin with, we observe that, for any and , the function is measurable in . If g is bounded and uniformly continuous in , this is clear. Indeed, as it has been recalled in Section 2, the function is continuous in for any . Hence, taking (1.3) into account and adding and subtracting , we can estimate
for any , and the last side of the previous chain of inequalities vanishes as tends to .
If the function g is as in the statement of the proposition, then we can approximate it by a sequence of bounded and uniformly continuous functions in which converge to g pointwise in and satisfy for any .1 Since the sequence is bounded and pointwise converges to g in , by [9, Proposition 3.1 (i)] converges to as pointwise in . Hence, the function is measurable in .
Proving that z is continuous in is an easy task, based on estimate (1.3) and the dominated convergence theorem. Hence, the details are omitted.
Fix . The first estimate in (A.5) with and the assumptions on g allow to differentiate z with respect to (), under the integral sign, and obtain that is locally θ-Hölder continuous in , uniformly with respect to , and
To conclude that is continuous in , it suffices to prove that, for any , the function is continuous in . For this purpose, we apply an interpolation argument. We fix such that . Applying the well-known interpolation estimate
with and , from the continuity of z in and the local boundedness in of the function , we conclude that the function is continuous in . Hence, . Estimate (A.2) follows from (A.6). Further, estimate (2.4) and the assumption on g imply that
for any , whence the second estimate in (A.1) follows at once.
Let us now assume that for any . Arguing as above and taking the second estimate in (A.5) with (resp. ) into account, we can show that (resp. ) for any and (A.3) (resp. (A.4)) holds true. Applying the interpolation inequality
with we deduce that the second-order spatial derivatives of z are continuous in and, hence, in due to the arbitrariness of .
Finally, to prove the differentiability of z, we introduce the sequence , where
As it is immediately seen, converges to z, locally uniformly in and each function is differentiable in with respect to t and
for such values of . Since for any , and converges to locally uniformly in , by [9, Proposition 3.6] and the dominated convergence theorem, we conclude that converges locally uniformly in to . Thus, we conclude that z is continuously differentiable in and, therein, . ∎
Let J be an interval and let be such that is bounded in for any . Then the function is measurable in J.
To begin with, we observe that for any the function is continuous in J. This is a straightforward consequence of the uniform continuity of g in for any bounded interval compactly embedded into J. To complete the proof, it suffices to show that converges to for any . Clearly, for any fixed , the sequence is increasing and is bounded from above by . To prove that converges to , we fix a sequence such that tends to as . For any , let be such that . Without loss of generality, we can assume that the sequence is increasing. Then for any . Hence, the sequence converges to and this is enough to conclude that the whole sequence converges to as . ∎
Finally, we prove some interior -estimates.
Let be a bounded open set and let solve the equation in . Then, for any and any radius such that , there exists a positive constant such that
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 such that and a sequence of cut-off functions such that and for any and , where
Since the function solves the equation in , where
we can write
where is the evolution operator associated to the realization of the operator in with homogeneous Dirichlet boundary conditions. It is well known that
for any , and . Since for any , from (A.7) we obtain
Now, for any we set and estimate the function under the integral sign. At first, we note that
By interpolation and using Young’s inequalities we obtain, for any ,
Collecting the above estimates together, we get
Now we fix and . Multiplying both the sides of the previous inequality by and summing up from 0 to N yields
Since is bounded, taking the limit as on the left-hand side of (A.8) we conclude that
for any . An interpolation argument gives for any , and this completes the proof. ∎
L. Angiuli and L. Lorenzi, Compactness and invariance properties of evolution operators associated to Kolmogorov operators with unbounded coefficients, J. Math. Anal. Appl. 379 (2011), 125–149. Web of ScienceCrossrefGoogle Scholar
L. Angiuli, L. Lorenzi and A. Lunardi, Hypercontractivity and asymptotic behaviour in nonautonomous Kolmogorov equations, Comm. Partial Differential Equations 28 (2013), 2049–2080. Google Scholar
G. Da Prato and M. Röckner, A note on evolution systems of measures for time-dependent stochastic differential equations, Seminar on Stochastic Analysis, Random Fields and Applications V, Progr. Probab. 59, Birkhäuser, Basel (2008), 115–122. Google Scholar
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer, Berlin, 1981. Google Scholar
M. Kunze, L. Lorenzi and A. Lunardi, Nonautonomous Kolmogorov parabolic equations with unbounded coefficients, Trans. Amer. Math. Soc. 362 (2010), 169–198. Google Scholar
L. Lorenzi, Analytical Methods for Kolmogorov Equations, 2nd ed., CRC Press, Boca Raton, 2017. Google Scholar
L. Lorenzi, A. Lunardi and A. Zamboni, Asymptotic behavior in time periodic parabolic problems with unbounded coefficients, J. Differential Equations 249 (2010), 3377–3418. Web of ScienceCrossrefGoogle Scholar
About the article
Published Online: 2017-02-04
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”.
Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 225–252, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2016-0166.
© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0